Sweep the basement to remove dirt and dust. Mop the floor and scrub the walls with soapy water to get them clean.
Let the basement dry out for several days. Concrete is very porous and it will take a while for it to get completely dry. If the outside is nice and you have basement windows, opening them will help speed the process, as will turning on a few box fans.
Seal hairline cracks with a waterproof caulk. Squeeze the caulk tube as you go along the crack. Go back and flatten the bead of caulk with your finger. Also caulk around windows.
Paint the floor and walls with a water sealant concrete paint. This won't protect against condensation but will help with water seeping in through the concrete. Use paintbrushes for rough block concrete so that you can get into all the little dips. Also use paintbrushes for the trim. Use a paint roller for smooth concrete surfaces such as the floor. Let the paint dry several days. Fans will help speed the process.
Attach rigid foam insulation board to the walls to create a vapor barrier against condensation. Squirt construction glue on the wall and let it sit for a few seconds before applying the foam board. You can cut the board to fit the wall, if needed, by scoring it with a knife. Snap the board to remove the excess. Also use the knife to cut out places in the board to work it around pipes or outlets.
Press the board to the wall and hold it for a few seconds to help it set.
Caulk the seams of boards that butt up against each other as well as along the top and bottom.