It has been over 90% humidity some days recently ... like walking through a cloud some days (a rather hot cloud). If the hay isn't molded, you should be able to rescue it, though. You'll want to make sure it's not in anything that's going to trap the moisture (so no plastic bags, etc.), and you'll want to get it spread out to help drying and avoid moisture trapped in pockets formed by clumping of the hay. Spread it out as best you can in the sun somewhere it can dry out. It's supposed to be sunny (and hot) and much less humid tomorrow and Saturday, so that'll be a good opportunity for drying hay (though terrible for my garden ...). After it's dry, you should be store it normally, though it may have a reduced shelf life. Storing it in multiple containers isn't a bad idea, so if some goes moldy, it doesn't take out all of it (better to toss $20 than $50, if you have to).
Basement may or may not be good for storage; it depends on your basement. If your basement is dry, it could be very appropriate (think root cellar); if your basement tends to be damp or musty, it's not a good idea (and you might consider investing in a dehumidifier). We've stored bags of litter (before switching to fleece) and hay in our basement before, and it's been just fine. Our basement is dry, though, and we run a dehumidifier in the summer and a fan year-round to keep air circulating. Basements are at least usually a bit cooler, as being partially underground helps keep them more thermally stable and moderated (warmer in cold, cooler in heat). Ours usually sits around 60° F.
