Look for sources of moisture in your basement. Some basements, especially in the summertime, will be damp from condensation because the air is warm and humid while the basement walls and the soil outside of them are cool. More severely damp basements are often caused by leaks from cracked or porous walls, poorly sealed windows or faulty plumbing. Check walls and window frames thoroughly and pay extra attention to areas underneath the kitchen and any bathrooms. If you are able to find the sources of any major leaks, have them fixed as soon as possible.
Confirm that none of your home's gutter downspouts are draining against basement walls. If they are, redirect them away from the walls. If the land outside slopes downward toward the foundation of your house, re-grade it so that it slopes away. This will eliminate the problem of water pooling against the basement walls and possibly leaking inside.
Seal off your basement as much as possible. Keep all doors and windows to the basement closed to prevent warm air from coming in and creating condensation. If your basement has a dirt floor, keep it covered with plastic sheeting so that no moisture is able to come up through the soil. If you have a clothes dryer in the basement, check the vent duct to confirm that it's securely attached to the dryer and properly venting air outdoors.
Apply a waterproof sealant to concrete and masonry walls in your basement. These sealants are available at any home-improvement store; they are applied with a regular paintbrush and work to seal the pores in concrete and bricks to prevent any outside moisture from seeping into the basement through the walls.
Place a dehumidifier in the basement after sealing off all known sources of moisture. Dehumidifiers remove moisture from the environment by circulating the air over refrigerated coils and collecting the resulting condensation in a small water tank, which you will have to empty periodically. Once you have gotten the moisture level in your basement under control, the preventative measures you have taken should keep the problem from returning.