Make sure the shower is completely and totally dry. If you have another bathroom, use only that one for about a week so that every trace of moisture dries up. If you can stop using the bathroom in question for long enough, you may want to remove the caulk, disinfect the area, and then let it dry for a good week before putting the new caulking in to place. If you do it too soon and there's still moisture there, you'll only be seeing the mold return.
When you're running the caulk around the seam, the back of a small spoon does a pretty good job of pressing it in to the corners.
