I have this same issue, since my reloading bench is in an unheated/unACed detached garage.
A few things to watch out for. Keep a GOOD coat of BreakFree or some other rust preventative in the steel parts of your press when not actually reloading. I bought a brand new Hornady LNL AP press a few years ago, set it up and went to work. Took a break (which unintentionally turned out to be a few days), came back and all the steel parts were coated with rust. Came off easy with BreakFree and a little steel wool, but it was definately panic time when I first saw it. The humidity and fingerprints sure did a job on it! Now, I keep a coat of BreakFree on the steel parts and haven't had a single rust problem. Just wipe the BreakFree off, load, then use something like a plumbing acid brush to coat the parts again.
During the winter, even if the air is pretty dry, watch out for flash rust if you start a heater up in your reloading area. When I was younger and living with my parents, I had a press set up in their barn, which was also unheated/unACed. Turned on a propane heater in the dead of winter and worked on sorting brass for a while till the area warmed up. I found flash rust forming in just 30 or so minutes because the cold steel "sweated" when exposed to the hot air coming off of the propane heater because of the temp differential. I moved the heater to a different spot, so that the hot air wasn't blowing directly on the press and it could heat up slowly as the air heated up and never had any trouble again.
I have heard stories about dead primers when reloading in extremely humid conditions from the moisture penetrating the priming compound before the primers can be seated, the powder dropped and the bullet seated. I've never had any trouble with this, so I don't know if it is true or not, but it's something to watch out for in extremely humid conditions.
Make sure that you keep your powder and primers stored in the house itself, not in the basement. Powder can get damp and clump and, I suppose, the primers can become damp and dead in extremely humid conditions if left there long enough. I accidentally left a cardboard can of powder in the barn at my parent's house all summer one time and found that the powder had become damp and clumped. When I took the powder and primers in the climate controlled house with me when I was done, I never had any trouble at all.
Easiest thing to do would be to see if you can put a heat/AC duct (or more, depending on the size of the basement) into the basement. This would take care of the heat and AC problem and would also likely take care of a lot of the moisture problems with the AC running. I don't know how feasible this is for you, but it's something to consider. Even running a fan would help out a lot, since it would keep the air moving. If the basement is really damp, here's something that can help. Get some pure calcium chloride (used as an ice melter and as an additive in concrete during the winter, should be available at home improvement/hardware stores) and put a cup full or so in an old sock, the amount isn't important at all, just leave enough room to tie off the sock and hang it. Tie the sock off and hang the sock over a bucket. The calcium chloride will absorb the moisture in the air and drip into the sock. When the bucket starts getting full, dump it and start again. When the sock starts getting empty, refill it and hang again. This method works REAL WELL for damp/musty basements. My dad did this for years, before they got central air in their house, and it works every time. Just remember to get PURE calcium chloride for this purpose, not one of the mixed/blended ice melters.
Good luck with your setup. Just take steps to control the moisture and you should be OK in the basement.
Bub