Basement It's not wet just spots on concrete
Basically what appears to be happening is that ground water is saturating the concrete, which is acting like a sponge. The concrete will continue to absorb the ground water until that situation is remediated.

Some solutions to this include digging down to the footing of your foundation completely around your entire house, and installing a french drain to direct the water away from your foundation. The foundation walls should also be waterproofed from the outside. This is typically done by using attaching a combination of tar, plastic sheets, and drain tile to the exterior foundation walls that are below grade.

You may need to cut into the basement slab, and run drains under the slab and into a sump pit if the source of the water is coming from below the house. If the walls of the foundation on the inside are bone dry, but the slab is soaking wet, then you pretty much know that it is coming from below.

Obviously, both of those solutions are expensive, and are typically above and beyond a DIY project, unless you have construction experience, and you know exactly what you are doing.

One thing you can do to partially remediate this problem on a budget would be to seal the surface of the concrete so water cannot weep into your living space. 

Epoxy floor paint may be an option as long as the slab can be dried out. The floor needs to be nearly bone dry before you can apply the paint. If it is too damp, the paint will not bond properly to the concrete, and it will cause the paint to blister, and peel. I had a similar problem in my basement, about 4 years ago, and the epoxy paint solved the problem. So far it has held up pretty good. I will need to touch it up in spots from normal wear and tear, but it has not peeled at all.

You can try drying out the floor by using a combination of a blower fan, and dehumidifiers. You can also try laying down a thin layer of clay based kitty litter (unscented is preferable) to soak up excess moisture. If you can keep the floor dry for at least a day, then you have a good chance that the epoxy will take. If the floor never dries out, then you can wait and try again when conditions are favorable. The best time to do this is during colder/dryer months where the relative humidity is at its lowest.

When you apply the paint, make sure you read and follow the instructions exactly. You would typically apply a degreaser/etching solution first, followed by a concrete primer and bonding agent, and the finally the epoxy paint itself. If all goes well, then your basement shouldn't have any more water seepage issues.