If a runner is stealing the second base, and the catcher drops a third strike is first base considered occupied or can the batter run to first and be safe?
According to the MLB rules 5.06(a), a runner occupies a base when:
Occupying the Base
(1) A runner acquires the right to an unoccupied base when he touches it before he is out. He is then entitled to it until he is put out, or forced to vacate it for another runner legally entitled to that base.
So in this case, if the runner had reached second base before the ball reached the plate (and thus became a "dropped third strike"), the batter would become a runner. If the runner was still on the way to second base, the batter would be called out.
In most instances, the runner will not have reached second base prior to the pitch crossing the plate, and thus the batter will be simply called out (unless the runner left while the pitcher was still holding the ball and nobody paid attention; this happens sometimes when a team is down by multiple runs late in the game and the runner on first is not the tying or winning run).