tl,dr: The short answer is no, with caveats.
The longer answer starts with "well, you can kill your car with anything if you try hard enough."
Let's use specific examples of when you should use your transmission for engine braking: Waimea Canyon or Mount Washington. In either case, you are descending thousands of vertical feet at significant grades. If you were to ride your brakes the entire way down, it's highly likely that the brakes would overheat and become less and less effective until they were essentially useless.
This would be followed by a loud crashing sound.
In fact, at the Mount Washington Auto Road, they will not let you proceed up or down the hill unless you are in "2" on an automatic transmission vehicle.
Caveat: when you intentionally put your car in a lower gear and drive down a steep grade, you are giving gravity control of your engine revs. As I say, it is sometimes important to use engine braking (we've discussed the benefits to fuel economyfuel economy before). However, you are the driver of the vehicle so you have to make the call about how much is too much.
A simple rule of thumb: the redline is still the redline. If your engine braking is pushing the limits of the rated engine speed, it's time to use the brake pedal and slow down significantly and, possibly, drop down another gear. Otherwise, you're going to break your engine in an effort to save your brake pads.