My check engine light came on and the mechanic said the problem is that the secondary air injection pump motor broke. It's a used car (2004 VW new beetle) with about 150,000 miles on it and the parts and labor would be over $700. Is it worth it or even necessary to get this replaced?
Yes, it will probably be necessary to fix the problem in order to pass emissions requirements.
The secondary air injection pump on the Beetle is apparently used to accelerate the transition from cold start to hot catalytic converter operation. Without this system, it will take longer for the catalytic converter to reach operating temperature.
The emissions regulations vary by location, of course. Where I live, one of the factors that is tested is that time of transition from a cold cat to a hot (much more efficient) cat. I doubt that your vehicle would pass without the assistance of the pump.
As always, it's your car and your money. You have to decide whether it's worth spending one on the other.
Some cars, the air injection pump is an electrical pump. Check the fuses. If you remove pump, your computer will still look for it. You will need to fool the computer as well.
Then there is the part about federal (and typically local) emissions laws. Its illegal to remove any emission control equipment and doubly illegal for a professional mechanic to do it.
Then there is re-sale value. Pretty much impossible to sell a car with missing emissions controls.
Typically air-pumps only run for the first couple of minutes until the catalytic converter heats up. On some cars, if you idle too long, the cats cool down to the point they are not effective. So the air pump turns on again.