Hi there!
Ah yes, the joys of humidity!
It drives me insane, my gorgeous little anvil started rusting less than 24 hours after I first used it!!!
Steel wool would probably only work for the tiniest amount of rust. I had a little more, so I used some wet/dry metal sandpaper that I got from my hardware store. I started with a coarser grade and then worked my way down to the finer ones, last of all I then rubbed the anvil with some 000 steel wool and soap. It left me with a lovely rust free surface
That being said I didn't have a huge amount of rust to deal with, larger amounts might require a 'hard core' approach. I now keep my little anvil wrapped in an oily rag and stored in some ziplock bags.
I'm no expert though, so if anyone has a better way, I'd love to hear it!!!
Niquita