We have a constant drip of water leaking from a pipe in our condo garage. There is a crack in the cement floor near the drip. Will the drip cause damage to the garage cement floor?

It could if it freezes and expands. It can also allow algae and mold to build up in the crack as well as seep into the ground and soften the dirt under the slab or footing. Plumbing is usually covered by the condo association. Can you see what the pipe is connected to? Hot water heater TPR, Condensate drain?