During the curling season, it is fine (and even recommended) to leave them on the ice when not in use. If you do need to remove them from the ice for security reasons, you should keep them in a cold location, as close to ice temperature as possible. Some commercial establishments and schools build a "rock shed" with a small ice pad in the bottom for storing rocks securely when not in use.
When storing over the summer, any dry location is fine. It is best to store rocks during the off season upside down on their handles to avoid scratching up the bottoms.
If you find that the bottom surface of your rocks tarnishes, you can use a hand sander and extra fine sand paper to restore the surface.