Wood is used on the outside of houses in all kinds of weather, so I think it could work in a bathroom.

Wood is fine, so long as it's protected. Talk to the guy at the paint store about which paint is best (water-based, oil-based, enamelâ¦). He'll also tell you to have the anti-mold and mildew additive mixed into the paint. Benjamin Moore actually has a "Kitchen and Bath" paint, which has the anti-mold/mildew stuff in it.

Presumably you also have an exhaust system in your bathroom? Use it when the shower is going to exhaust steam.

You'll be fine.