I keep my strops in the bathroom. It gets steamy. I've seen no ill effects. However, I also keep tabs on my strops and make sure they're properly oiled. Oil and water don't get along well, so a properly oiled strop (front AND back) is going to less susceptible to damage from steamy shower air. I suppose (though he's no longer around to ask) that's why grandpa's father's strop is still in good condition after hanging in regularly steamy bathrooms for ages.

Peace,
Jim