Diagnose the problem
If you have general dampness most of the year, your home probably sits in poor-draining soil (clay), which is damp most of the time. Normally, damp-proofing the exterior of the foundation and basement slab prevents this problem, but if your builder didn't do this (it wasn't commonly done on older homes), your simplest and least expensive solution is to buy a humidifier to control the dampness.
However, if the dampness is more episodic, like after a rain, or wet areas regularly appear along walls or floor cracks, there's probably significant water pressure against your basement walls. Since grading and adding downspouts hasn't worked, installing a drainage system under your concrete slab may be the most effective long-term solution. This drainage system relieves the water pressure. It consists of a corrugated plastic pipe (called drain tile) that collects water before it enters your basement, then channels it to a sump pit where a pump discharges it away from the house. The drawback is the cost. It's expensive and usually requires professional installation. Expect to pay at least $25 to $35 per ft. of tile.
Consult a waterproofing contractor to find out if you need this system or if a lower-cost option will work. If groundwater is seeping in under the floor (which often occurs because of a high water table), there's really no other permanent fix. The contractor will cut out and remove the perimeter of the concrete floor for the drain tile, dig a trench and embed it with gravel, and then dig a sump pit. To finish the job, the contractor will install drain tile and place dimpled plastic sheeting at the bottom of the wall and under the new section of concrete floor to channel water entering through the wall to the drain tile, then patch the floor with concrete.
Basement dampness isn't usually a problem in new homes. The builders typically have installed drain tile around the outside of the foundation and have damp-proofed basement or crawlspace exterior walls before backfilling. This can also be done for existing homes, but it requires digging along the outside of the walls all the way down to the footing. It will destroy all your foundation plantings— shrubs, flowers, etc.—and is expensive, which is why we recommend putting it under the slab for homes already built.
If water enters your basement only through cracks in the concrete or masonry walls, but not through the floor, you have another option. You can install a “drainage channel” along the wall on top of the floor (for concrete walls) or on top of the footing (for masonry walls). The channel collects the water and drains it into the sump pit. While this method is less expensive than installing drain tile, it's not as effective since it deals with the water after it's in the basement.