The Answer:

It's a popular belief, but it isn't true.

In a 1981 column, Cecil Adams calculates that under ideal conditions, a penny dropped off the Empire State Building would reach a speed of about 280 feet per minute. By comparison, a handgun bullet of comparable size typically travels at more than 800 feet per minute. Add wind resistance, the effects of tumbling, and that a penny isn't shaped in such a way as to be an effective projectile, and it becomes clear that falling pennies are not particularly dangerous, let alone life-threatening.

This has also been explored on the fourth episode of Mythbusters, in 2003: the hosts fired pennies from a special gun simulating the speed a falling penny would reach. It turns out not to be able to even break human skin.

—The Fact Monster