I have mostly the "cheapie" cutters, the silver metal ones, and they can rust if you leave them wet for too long.
I wash mine in hot soapy water by hand, using the large wilton tip cleaning brush to get into the small spaces, rinse well, towel dry, and then put them in a warm oven to make sure they're completely dry before I put them away.
Mine are mostly stored in an old wooden chest of drawers with shallow drawers. I wouldn't store them in an airtight container, just in case any moisture gets in, so they won't rust.
hth!
Laura.
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