Depends on how moist we're talking. Is it wet all the time? How wet? Is there standing water? Is the floor and/or walls wet to the touch? Does it EVER flood? Or are we just talking a general "stuffy humid" feel to it?
If it's wet wet, you're not going to want to spend the kind of money it would take to make it suitable for chinchillas. It'd require a french drain and/or a sump pump.
NOW, if it's just a little wet...a dehumidifier will work. I prefer the type that hose into a drain but if your basement doesn't have a drain then you'll need one that has a bucket for emptying. You can buy ones that have both and choose which way you drain.