How do I keep water from getting under the garage door?
@ArchonOSX's answer is good, as long as the local drainage is good enough. If not, I'd install a trench drain that extends further down the slope to where it can discharge back onto the sidewalk, but that depends on the situation.

I'm guessing the door is 8' wide, and the sidewalk falls 5" across it, which means the sidewalk has about 1/2" per foot pitch.

Let's say you install a level trench in front of the door, starting 1" below the door and with a depth of 4" (with a grate on top). At the downhill end of the door the trench bottom will be 4" below the sidewalk.

Then you have to get rid of the water. Drains should have 1/8" per foot pitch, so the sidewalk will fall 3/8" per foot faster than the drain. So, it will take another 11 feet before the sidewalk is below the bottom of the drain. If you have that kind of space you can just extend the trench to that point and then turn it back to the sidewalk.

If not, improvise.