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Are classic cars safe? I am interested to know the differences between modern vehicles and vehicles of old from the 1960s.
Physical safety
Modern cars are amazingly more safe than classic cars. Guys that are into classic cars frequently throw around phrases like "They don't make them like they used to!" or "This is built like a tank with real American Steel!", but when you look at a classic car in an accident, the results are pretty obvious.
In 2009 this crash test was done between a 1959 Chevy Bel Air and a 2009 Chevy Malibu.
Click for video
Source: http://www.iihs.org/iihs/sr/statusreport/article/44/9/2
The aftermath of the crash shows how the modern "crumple zones" almost completely protect the driver's area in the 2009 whereas the 1959 driver would certainly be badly crushed.
In addition to crumple zones built into the frame, there are other thoughtful features like collapsible steering columns and as a high tech option the car will call and report the accident to emergency responders for you.
From the comments (thanks tallpaul): here is another video of a 1980 Volvo and a 2000 Renault. Not classic per se but it does show a marked difference in technology even in that 20 year span.
Anti-lock Braking Systems
Modern cars also come with ABS which reduces stopping distances while maintaining a level of steering control. The difference between the car sliding to an uncontrolled stop and quickly slowing down while being able to steer around obstacles or even to just stay on the road is huge.
Restraint Systems
Cars from the 60s and earlier don't even have a 3-point seatbelt (shoulder belts), but modern cars are required to have them, and many also have seatbelt tensioning systems that tighten the belt and hold you in the seat in an emergency.
In addition to better belts modern cars also have several airbags to cushion the occupants in an accident.
No, they are not safe
Safety standards dating back to before now were not as stringent as they are now.
The further in time you go back the less safe they become.
Safety has been driven by governments and as regulations have become more stringent over time car manufacturers have had the responsibility to conform to the compliance stack of the time. Whether it was driven by regulation or legislative action.
A 'classic' car from the 50's won't have.
break away motor mounts
break away stearing column
air bags from front impact to passenger curtain
seat belt impact tensioners
reinforced side bars in the doors
anti-lock brakes
As well, proactive crash prevention measures and technologies have been developed to create awareness into a critical situation before it becomes a disaster.
tire pressure monitoring
blind spot detection
adaptive cruise control
Lane-departure warning/wake-you-up safety
Emergency brake assist/collision mitigation
Rearview camera's
All of these technologies as well as many not listed have contributed to higher survival rates in accidents over time.
Conclusion
Modern cars are more safe. Classic cars are less safe than modern cars. Be safe.
You've already seen the safety comparisons.
With that being said, classic cars are for fun. If you're looking for safety, a classic car isn't for you. If you're looking to have fun, go for it.
Like most things in life, there's a balance that you have to evaluate. No one can answer that for you. You have to do it for yourself.
Are you willing to take the gamble because you really love the car? If so, go for it. But if you value safety over having a fun daily driver, go with a new car. You have to decide where your priorities lie. Personally, I drive a 2006 Nissan Frontier, mainly because I can't afford a '65 Ford Falcon.
At almost 40 years old, I still ride motorcycles and want a classic car. That's where my heart lies. No one can tell you where your heart lies. That's up to you.
Classic cars are significantly less safe than modern cars. In a classic car, it is both harder to avoid a crash and more likely that you will sustain serious or fatal injuries in the event of a crash. It's the former point I'd like to emphasize in this answer.
First, a classic car will not have features like ABS, traction control, or stability control. This means that it is significantly more difficult to maintain control of the vehicle under adverse conditions, making a crash much more likely to happen. Even if you're an experienced and very safe driver, conditions beyond your control, like a deer running across the road with limited visibility or an out-of-control vehicle veering into your path, can force you to make evasive maneuvers that are much more difficult to succeed on without these safety features.
Second, as others have mentioned, modern vehicles are designed to absorb and divert crash forces away from occupants; classic cars are typically not designed to do this and will transfer much more of the impact to the driver and passengers. @JPhi1618's answer demonstrates this well so I will leave it at that.
You're asking two different questions - are they safe, and are they as safe as a modern car.
For the second question - No. An older car without all the modern safety features will not protect you, your passengers, or pedestrians as well as a modern car will in the event of a crash - You don't have airbags, crumple zones, ABS, NCAP ratings and so on.
For the first though, it really depends on what you mean by safe - If you're driving a classic car, you won't generally be driving as fast, and you'll be more in tune with your surroundings - you have fewer driver aids, forcing you to concentrate more than the 'average' driver. This hopefully makes you less likely to cause an accident - but of course doesn't insulate you from other people! Whether that makes it safe to drive a classic as your daily driver depends on many factors - I would never do so in a heavy city commute, but would happily do so if I were just pottering around country lanes.
Of course, other opinions may vary!
Given the assertion that most car accidents occur at speeds of 12MPH or less, most classic cars should be considered safe. Your odds of surviving a crash at parking-lot speeds are very good. However, your chances of walking away with only some bruises are much lower than with a modern car. In even a walking-speed collision, a classic car is going to transfer much of the impact energy to you, due to the rigid frame and body construction. Couple that with a simple lap belt and full metal dashboards and you can see where morbid comments like "just replace the radiator and hose off the dash and sell it again" came from.
In the event of an accident, how does a classic vehicle compare to a modern machine?
Badly.
Are safety features on new vehicles really a life saver?
Yes.
Can anything be done to improve the safety of classic vehicles?
There are certainly safety improvements that can be made. You can fit better brakes and tyres. You can sometimes retrofit collapsible steering colums. You can fit better seat/restraint systems to better secure the occupants during a crash (it's geneally better in terms of peak decelleration experienced by your body during a crash to be tightly strapped in than to fly forward).
What you can't really do much about is the behaviour of the body/chassis during a crash with a hard object. Modern cars are designed so that the area in front of the people crumples while the area the people are enclosed in remains solid reducing the peak decelleration experienced by the occupants and making sure they don't get crushed by the collapsing car.
Racing style roll cages work for racers who are securely belted and helmeted but are not a great idea for road use.
Are classics safe enough to be used as a daily driver?
That really depends on your risk tolerance. People used to drive those cars all the time when they were new. Most of them survived but a significantly larger proportion than today got killed or seriously injured in accidents.
As others have said, you're asking different questions so I will address them separately.
Classic cars are safe to drive
Classic cars are simply older cars. They were driven successful by people of the time who didn't die in them and nothing has fundamentally altered the safety of the car since that time provided they have been properly maintained. If you drive a classic in the same way as you drive a modern car in daily use you still have a low lifetime chance of dying or injury. As always, you can decrease - but not eliminate - your risk by sensible, defensive, driving techniques.
Modern cars are much safer to drive
Consider this graph showing deaths per 100,000 people in the US (black line, right axis) and note the dramatic fall:
Then consider that this has come despite a large rise in the number of miles driven per person and the number of cars on the road. The major contributing factor to this is the many improvements to car safety: better brakes, crumple zones, air bags, anti-lock braking systems, lane warnings and so forth. So although classic cars are safe to drive, they are much less so than modern cars; it is - as always - up to you to assess the relative value that you put on the pleasure of driving a classic and the higher risk to your physical safety.
Safety in crashes (protecting you): Improvements are visible on a decadal scale and the tend to aggregate over time. (this is somewhat backed up by the plot with the dips discussed above)
2016: Cars commonly come with lane centering, adaptive cruise control, devices to keep the driver awake, and automatic stop features.
2010: Pretty much every car comes with ABS and ESC/Traction control. Cars have airbags all over the place and advanced collision cells and crumple zones
2000: ABS and Dual airbags are standard in many new cars. Middle passengers likely have a 3 point restraint. Child seat anchors. Passenger head restraints.
1990: Airbags become available. All cars have crumple zones.
1980: Fuel tanks have to be inside the car's frame.
1970: 3 Point seatbelts are starting to be commmon
1965: Unsafe at Any Speed Published (from here you can add things like "roofs that don't collapse when the car rolls)
Things that help the driver be safe
2016: Rear back up cameras are ubiquitous
2010: Rear warning systems common. SUVs built on car platforms handle well and resist rolling. ESC and traction control are ubiquitous.
2000: Passenger cars commonly come with 4 wheel disc brakes, wide tires, and abs: better handling and shorter stopping distance.
1990: Cars come with both driver and passenger side mirrors. Finding a car with rear wheel drive is rare: average new car has superior cold-weather road handling.
1980: Finding a car without power steering is rare.
1970: Cars commonly come with front disc brakes, stopping distance decreases dramatically.
There are a few areas where safety has decreased over time:
Can anything be done to improve the safety of classic vehicles?
You can attach & use safety equipment like a racing seat & harness, helmet, even a roll cage: 
This article about Racing Safety Equipment has lots of info, including a warning about roll cages and helmets (especially when putting steel bars near your head):
An accident involving a roll cage and a driver without a helmet will never end well
This site (http://www.cuscousainc.com/products/roll-cage.html) sells roll cages (apparently pre-fabricated) for several models, including it looks like 5 passenger vehicles: 
Are classics safe enough to be used as a daily driver?
If you drive a short distance at low speeds on generally safe streets, and look both ways before crossing a street (even when you have a green light), then you're probably safe enough. Compared to a bicycle / motorbike, almost anything that puts a door between you and other vehicles is probably an improvement.
It may depend more on your idea of "safe enough." I've read that the European idea of a safe car is one that's fast and nimble enough to avoid an accident, while the American idea is to wrap yourself in as much steel & mass as possible, and let the laws of physics do their job.
No car is safe! It can (and does) hurt and kill much more than anything most of us do frequently. Best safety equipment is the operator of the vehicle....NOTHING can replace an alert, smart, and knowledgeable driver! Anyone with knowledge of their vehicle and its limitations can be a safe and courteous fellow driver.
With that stated, the advancements in newer cars are quite astounding. They have many technologies that will help protect the occupants. However, a loose 2x4 can slam through a modern windshield/window as effectively as it can in classics.
Also, with each item that a modern car will now 'do for you', most drivers become dependent upon the equipment and do not know what to do if a real road emergency arises. Anti-lock brakes are a great example. They are wonderful and will help in emergency braking. However, this gain has also almost eliminated the casual driver from having the experience on how to handle a skidding car! I have gotten myself out of several troubles by simply never getting my car into a bad situation in the first place!
However, I gladly put my family in a safer, newer vehicle every chance I get!
Know the risks...know the vehicle...be smart...happy motoring!
in response to the original question: "are modern cars safer then classic cars"
(firefighter here) simple answer: Yes. A lot.
why?
From my own experience as a firefighter (in the road rescue squad), this seems to be the main differences. Accidents still happen and car crumbles, but there is a huge difference between old and new cars (as well as cheap and "expensive" ones).
This were only the "damage-reduction" systems. I have not mentioned all the automatic brake systems, ABS etc which help to prevent accidents
Just watch some videos of car-crash tests. You will see the difference;)
If you are after more detailed answers, please read the other answers or ask in the comment
I have been involved in several aspects of the old car world since the early 70's. This subject has come up more and more in the last maybe, ten years and I have spent alot of time pondering it and my involvement in historic racing has given me additional perspective on the subject I think. The basic answer is no, older cars are not safe as newer stuff for a few really important reasons. 1st, older cars (pre 69 ) are full of sharp edges in the interior - a death trap in an accident if you are thrown around inside. The use, in period, of lap belts cause their own problems. the seat can fold forward in certain kinds of impacts and serious injuries - spinal or simply face into steering wheel can result. The installation of fixed or locking seat backs with shoulder harnesses can help but not easily engineered on most older cars. Braking distance can be significantly longer in old cars. here in LA where folks drive inches from the car in front doesn't help much. You might say "leave more room in front" ...I have done this and other drivers just nip into the big space that you have left for them. Another issue that I have noticed as a racer is that the curb weight of modern cars is in some cases 50% more than old ones (no, not Lincolns and Cadillacs but smaller sporty cars) a modern Mazda Miata weights more than a Ford Falcon. This changes your bargaining power in just about any collision if you are the person in the old car. Well, that's my short list!
This is my experience with old classic cars. My father and I (he's passed on) rebuilt 9 old mopars, from a 1936 Chrysler to a 68 Roadrunner 383 Commando (as well as a 68 Satellite into a RR reproduction with a 440 and a 63 Dodge Polara Race car). No, Older cars are not as safe as newer ones and that has been well covered. However their are things you should do to make older cars safer if you want to drive them on the streets. First is the tires. Get rid of the old Radial Tires. Their Dangerous. If you have never driven with them they are very different than steel belted tires. Next you should do updates. The front end and brakes are the two biggest. The front end will help with keeping the car on the road (those old front ends were known for curvy country lane driving) and the brakes because you want to shorten the stopping distance. From their it's usually little things like shocks and better head lights. Remember, those parts were "cutting" edge in the 60's, we have better now. If you do these things, then yes that old car can be a daily driver.
