I own a small home built in 1960. There are no exhaust fans in the bathrooms. I would like to put up crown moulding and would like to know if there is a certain type I should use. Will the crown moulding swell?
Your concern about moisture is well-founded, but you should worry less about damage to the molding and more about damage to the bathroom itself. The fact that your bathroom is not vented puts you at risk for excessive humidity in the bathroom, leading to mold growth and unhealthy conditions, even rot. You need to figure out what it would take to get a bath fan installed. It must be properly vented to the outside of the house so warm, humid air does not build up in another part of the house, such as the attic.
In terms of the molding, choose a synthetic material such as Easy Crown - not a natural material like wood.