I had a similar problem with a house I rented last year - and in fact it was a huge leak.....my slumlord refused to hire a plumber. If you own your home, you may need to get behind the tiles.

Here's a quick, easy temporary fix. I would use a damp sponge to wipe off the damp caulk. Use a blow dryer to speed up the drying process both before and after reapplying. Don't use the shower for a few days - steam may make the problem reappear. The standard cheap white caulk may not be what you need.....I think I used a caulking product made by Elmer's.

Good Luck!