Assess the severity of your wetness problem. A damp basement with no visible water can probably be dried out with simple fans and the opening of windows, whereas a more severe problem will require a dehumidifier. Large amounts of standing water requires the use of a pump along with the dehumidifier. Smaller amounts of water can be removed using a Shop-Vac.
Install the dehumidifier if you have small puddles of water or fans were not enough to solve the problem. Follow packaging installation instructions carefully.
Install a pump meant for removing water if the previous steps are inadequate or if you have a major water problem. You may also wish to consider calling a professional at this time. If the problem is recurring, a sump pit and pump might need to be installed. Your local home improvement store carries pumps meant for removing small amounts of standing water from homes. Ask staff there for a recommendation based on your individual situation.
Dry out any movable floor coverings, personal items and furniture by putting things into the clothes dryer, hanging them out in the sun or spreading them out onto a dry area of floor in another room. The garage is a good place for this if you don't want wet things all over the house.
Replace any drywall that got wet in the case of flooding. Most experts recommend total replacement of water-contacted drywall to prevent major mold problems. Remove wet drywall and allow the wall studs to dry for at least a week in a dehumidified environment before reinstalling drywall.
Locate and repair the problem that led to the wet basement to begin with. In the case of flooding, install a sump pump or other moisture-activated pump. In the case of leaks, locate and repair the leak. In the case of general dampness in the basement, call a contractor to assess the situation. You may have a simple condensation issue, or you may have major grading issues outside. Visit the link in the Resources section to learn more about how to locate and repair common basement dampness issues.