Mount mop and broom holders on the back of the closet door. (Try the Crawford Spring Grip; $4, acehardware.com.) They'll be out of the way and can air-dry. Even better, for ventilation, insert a screen panel on the lower half of the door.
Leadrise® Mop and Broom Holder,5 Position with 6 Hooks Garage Storage Holds up to 11 Tools for Broom Holders, Garage Storage Systems Broom Organizer for Garage Shelving
Home- It Mop and Broom Holder, 5 position with 6 hooks garage storage Holds up to 11 Tools, storage solutions for broom holders, garage storage systems broom organizer for garage shelving ideas