We have a dehumidifier- also in the Midwest and our basement is already damp. If you want one, they're decently easy to find used I think.
That said, I have air-dried clothes for years on a rack without a dehumidifier, and nothing has ever happened. But usually I try to keep them in an area with decent air circulation, if anything because they'll take forever to dry otherwise. If you're worried, set the rack up in your bedroom and stick a fan in front of them. Or line-dry in the bathroom by hanging clothes off the shower when your roomie won't be home for a bit. I did this with roomies and they actually ended up joining me and pitching in for a rack- run it by your roomie, they may not care.