A damp or moist floor will quickly ruin any plans to refinish a basement into a comfortable and useful living space. Basement remodeling projects are often stymied by "water problems" especially when they are underestimated or not corrected before beginning other renovations. A wet basement isn't easy to correct and can sometimes be expensive, but remedial efforts now usually pay off in increased home values in the future.
Determine whether the "dampness" on the floor may be caused merely by humidity in the air. Some concrete basement floors seem to "sweat" moisture, especially during humid times of the year. Humid air may be condensing on a cool floor surface and causing this dampness. You can test this theory by setting up a dehumidifier in the basement and running it for several days after the sweating appears. If the floor dries up and seems to stay dry as long as the humidifier is on, it's possible that humidity is the extent of the problem and a constant-on humidifier is the easy solution.
Evaluate whether drainage problems around the house are causing water to seep into the basement and puddle or glaze the floor. If you notice the dampness or water only after a heavy rain (accompanied usually by standing water in lower elevations of the yard too), you have a drainage problem that must be corrected by major efforts. In most cases, drainage around your home's foundation must be repaired so gutter and drain water, and other runoff, flows away from the house quickly. No other treatment or method of repair really works to stop this kind of water seepage into a basement.
Attempt to stop minor seepage and other "sweating" on masonry walls and concrete floors with a special latex rubber product. This type of product can be brushed or rolled onto walls and floors, like paint, to provide a degree of "water proofing." Major water seepage problems will not be cured with the application of this product, however. Such products are amazingly odor-free (though not entirely odorless) but they are not attractive enough to be applied and left uncovered. Use these products under traditional floor and wall coverings.
Try to stop minor seepage or sweating on a concrete floor by using water-resistant masonry paint "sealer" products. Apply this paint-like product in at least two coats, and test if the dampness is abated. While this paint product often comes in several "decorator" colors, it's probably wiser to plan to subfloor the basement and use a traditional floor covering after you've sealed the concrete with this paint.
Solve some water problems by installing plastic water channels around the perimeter of the basement. This solution, usually installed by professional "basement de-watering" contractors, can work, but the channels are both expensive and sometimes unsightly and may interfere with other remodeling plans like the construction of new sheetrock walls.
Frame over the original concrete floors only after solving any major seepage or sweating problems. Lay a barrier of plastic sheeting over the floor area, and install a proper and level floor-framing grid using treated lumber or metal studs and moisture-resistant subfloor materials.