Are you sure it is a "mold" that people worry about? Lots of things grow where it is damp, dark, and no moving air.
I suggest scrape it off, if you can. If it is just on the surface - great! Just scrape it off, clean with stuff like Pat says, and cover with oil-based Kilz.
If it is actually IN the drywall, you will have a much bigger job to remove & replace all the affected DW - and a little more just to be sure. So try to do a little discrete digging to be able to make an appropriate offer.
Either way, you need to address the moisture problem. My daughter's reno had that all-around musty smell. We could see signs of unsuccessful attempts to deal with it for years.
First, make sure there is no way for outside water to get in (roof, gutters, caulk, insulated windows). Start running a dehumidifier.
Use a waterproofing substance in the basement (google moisture mitigation systems, like Schonox)
Surely you already know this, but I'll say it anyway. Where there is moisture, there likely could be termites and/or wood rot - so inspect carefully.
Let us know if you buy it!