As you inspect your basement, you may discover three common culprits that compromise your comfort and safety.
Water damage and high humidity cause mold to grow within walls. If you find mold, remove it and locate the source of excess water. Often you can remove mold simply by applying a 50 percent solution of laundry bleach and water to the affected area. As you work, be sure to provide plenty of ventilation and wear protective gear: gloves (made of natural rubber, neoprene, nitrile, polyurethane, or PVC; ordinary household rubber gloves won't stand up to bleach), goggles without ventilation holes, and a good-fitting N-95 respirator available at many hardware stores and home centers.
A 50 percent solution of laundry bleach and water is actually more effective for mold removal than using full-strength bleach, which is hazardous to breathe and could result in regrowth of bleach-resistant mold strains. If mold growth is extensive, you may have to replace carpet, insulation, and other affected materials. Mold is present in some form in nearly all homes, so don't overreact if you discover some in your basement.
Have the mold analyzed by a professional to determine its type and health consequences. Check the telephone directory under "Environmental & Ecological Services" or "Air Quality Measuring Service." Your state's public health department also can provide information.