Not knowing where the water is coming from, hard to tell what type ofsupplier to use - tile andmaybe surround issues a Ceramic Tile contractor would solve (also leaks at connecting shower doors), Door seal leaks a glass or door contractor would normally handle though some plumbers do too, a leaking pipe in the wall a plumber.
However, since you drywall damage and possible wood rot behind that, and don't know what kind of contractors you REALLY need till it is torn into, I would say get a Remodeling - Bath and Kitchen contractor to do the investigtion, because he is equipped (with the use of subs) to handle any of the causes.
Your other alternative, which would be cheaper if you guess right but more expensive probably if you guess wrong, would be to guess the cause and contact the appropriate contractor type for that cause - but that does not take care of remedying any mold, or the wall repair.
Personally, if I had to choose a contractor type, I would assume it is probably a problem with the caulk around the shower door frame or a door not closing tightly because the frame is out of plumb, leaving a gap where water can get around the door. If that is the cause you would have wetness outside the doors on the wall and/or floor after showers. If choosing just one specialty contractor rather than a general contractor to look at it, I think I would go with a Ceramic Tile contractor to solve the problem, but then you would need at least a drywall contractor to replace the damaged drywall and a painter to repaint - so you can see why going with a general contractor specializing in baths is probably better from the get-go, because you definitely (unless the repair will be a DIY job) need different types of contractors to remedy the problem than you need to repair the damage.
