How can I protect the wall behind a closet from developing a mold problem?
Mold needs three (or really four) things to grow: Stagnant air, moisture/humidity, and dim or dark areas with organic 'food'. The solution you're looking for probably involves using a permanent waterproof barrier (either a paint product -- which would be an industrial coating -- or a foundation product) in order to largely remove the humidity from the equation. Those coatings would need to be installed by a professional. As a cheaper and DIY solution, you might also consider 6mil plastic vapor barrier on the back side of the wallboard.