Not to throw another bucket of mud on things, but I'd bet you a box of doughnuts that your water isn't just seeping in through the basement walls, it is coming up from the ground too, under the slab. There are several things you can do: Damp-proofing (i.e. what Rox is talking about), exterior french drain, interior french drain with sump pit(s).
The biggest problem you have is that the ground isn't going to dry out anytime soon and that any rain we do continue to get is going to end up in your basement. Thems is just the facts. But when we do dry out, the only way you are going to stop any of this from happening in the future is to do one, if not more, of the suggestions thrown out. The good thing is you have a strong back and a weak mind, so hard rock labor isn't going to effect ya much...
let me know if you need someone to supervise and tell you what to do, I am more than qualified to stand around, direct traffic and drink your beer.