Inspect your basement for any evidence of water damage or moisture seepage problems. If your walls are finished with sheetrock, the wet spots will be evident and look dark in color. If you have bare concrete walls, you may notice that the concrete feels damp or even looks wet.
Inspect the outside of your home to look for any evidence of drainage problems. If downspouts are not connected or gutters are clogged, water might not be diverted away from your house properly. Also, it's important that the land around your house is sloped away to promote proper water drainage. Cure these issues first before you do anything inside your basement.
Run a dehumidifier in your basement to remove any existing moisture. It's important to dry out your basement to stop any mold growth and get the rooms dry. It's also a good idea to run a dehumidifier on a regular basis just to keep things from rotting.
Purchase a sealant paint for interior walls and apply it according to the manufacturer instructions. This paint is specifically designed to keep water out of your basement. This may require two coats of application.
If these methods do not work, consult a professional contractor. You may need to install drain tile and a sump pump to prevent further water seepage. This method is much more costly, but also very effective in eliminating water and moisture problems in basements.