Hi Barbara, a good stain blocking primer and thorough paint job is a good place to start. If you do have actual mold spay it with hydrogen peroxide or vinegar first before painting.
Do you have a dehumidifier in your basement? That will help deter mold to begin with - but you'll have to clean the floor with a bleach solution, and try to keep the air drier down there.
You might also need to invest in a dehumidifier too.
Damp rid can capture the moisture in the air and help. https://amzn.to/2W0lguH I had a closet one time that seem to have a musty smell and someone told me about damp rid. I put one in there and could not believe how much water it captured in a few days. This will not solve your problem but it will help while you are trying to figure out the cause.
Hi Barbara - There are several ways to get rid of mold, but most importantly, you need to address the root cause to keep it from returning. This article focuses on the bathroom, but is an overall great article that covers methods of how to remove mold, if bleach is safe to use, and how to prevent it from coming back. Hope this helps! Hugs, Holly