Clean it up real good, making sure to get rid of any soap film, and let it copmletely dry. Then use clear silicone caulk in all the inside corners, where wall meets wall and wall meets tub. While you're at it, it doesn't hurt to caulk around the plumbing handles, spout, etc.
The trick is to cut just a tiny bit from the caulk tip, and apply a small bead. Keep a dry rag handy to wipe your finger, as you'll want to use your WET finger tip to smooth it down.
Make sure to finger smooth each line as soon as you apply it, and don't go back to try to re-smooth any of it, as silicone caulk forms a skin within minutes of applying it.
By the way, if there is mold developing I find X-14
the best product for removing it.