Kelly, yes you can cover your basement floor, using 2x4 treated lumber laid on it's side to create your joists, or "sleepers", as they are called, but they will have to be planed and/or shimmed to get a level surface to put your new floor on. I'd strongly suggest your new subflooring also be treated material. Before putting down the sleepers, I'd seal all cracks in the floor thoroughly, and I think I'd also coat the floor with a concrete floor paint, but not put down plastic or other total moisture barrier; I think you'd have mildew problems if you did this.
Personally, I'd advise caution in covering this floor, though, unless you're quite sure that it no longer has a moisture problem or movement. Once it's covered it's hard to see what's happening down there.