Once you have your safety mask and goggles on begin looking for any moisture sources in your basement: a leaky pipe, a leeching drain with soil exposed, old washer machine etc. Address these issues immediately. Seal any cracks in basement walls and floors, cover any exposed soil with concrete, and fix leaks yourself or by a plumber. Continue wearing protective mask/goggles until all steps are completed.
Mold eats cellulose so remove any wall paper, carpeting, fabrics and insulation materials that have visible black/grey mold. Always check deeper into locations where you initially see mold. You may have to replace paneling and any underlying wall board/insulation that has been contaminated with mold. Non porous surfaces can be cleaned with mold/mildew remover and a scouring pad. Dispose of everything in a heavy duty plastic bags and remove it from your basement.
Invest in a dehumidifier. Once all of the mold contaminated materials are removed, using a dehumidifier is a huge step towards making sure mold doesn't return to your basement. According to the EPA a desirable moisture level to strive for is 60% or below relative humidity. Remember to empty out the water collector daily and this should finally get rid of mold in your basement and help keep it from returning.