3 basic choices
First, if you have central AC or AC in your gun room, issue solved. if you keep the room nice and dry, the safe environment will follow.
Second, one of the dessicant products. CheaperthanDirt.Com sells the ones i bought - like a quart size paint can, filled with little dessicant beads. If it EVER gets too wet, a couple hours in the oven resets it and you start over. I far prefer that over the salt -type ones at West Marine, which draw a residue of extremely salty water that has t be disposed of.
Third, one of the heater types. The ones i've seen look like little curling irons, you just need to be able to feed the cord out the door or through a hole in the safe. Didn't it have holes predrilled for mounting? If you didn't use them all, maybe you could feed the cord through and splice the plug back on it.
I HATE rust on guns. My Mossberg for some reason likes to develop a little rust on the field barrel (not, surprisingly, on the slug barrel) so after wiping and mopping it, i put in the can of dessicant - and problem solved.
good luck!
"Laws that forbid the carrying of arms ... make things worse for the assaulted and better for the assailants ... for an unarmed man may be attacked with greater confidence than an armed man." - Thomas Jefferson