First, I'd take a level and see if the tub is level both from front to back and side to side. If not, then there's probably no way to keep water from dripping out. It appears you are using a shower curtain, since there's no door there. On many designs, the panels snap together with a flange, so if the walls are straight (not common!) then it is waterproof (but still needs to be level).
Once drywall has become wet, it really should be replaced. If moisture has gotten back into the stud wall and floor behind, there could be a lot of rot. I think you should pull a bit off and look once you figure out the leak. Not knowing how yours went together, if the tub/wall joint is well caulked, it may just be running out the side. I think you'd have to be there to really tell.