Mold under Vinyl in Basement
So, first, test for seepage by taping a plastic sheet to the floor and waiting a few days. If you see moisture beading up under the plastic, you have seepage.
To stop the seepage, you can lay a 6 mil or thicker plastic moisture barrier down and then re-carpet or tile over it. Secure the plastic by laying glue both over and on top of it at the same spots: sandwiching the plastic between the glue to create solid, slip free spots/channels.
The best thing I thought of in lieu of a plastic moisture barrier was to paint the surface with a waterproof paint. However, most waterproof paints clarify that their product is not good for floors (UGL in particular: good for only walls). Some paint specialists with whom I consulted said that there is no solution; they had dealt with it before and there was nothing I could do.
I did not settle for that. The problem is that the waterproof paint is not resistent to foot traffic. Thus, my admittedly-untested-and-inexperienced solution was to just paint the floor with a waterproof paint, and then cover it with another floor convering. It might help to boost the paint with some mildicide additive, which is sold at most paint stores and at Home Depot and Lowe's.
Also, in one spot which would not receive foot traffic, I painted it with 2 coats of UGL oil-based waterproof paint boosted with mildicide, and then topped that paint with a mildew proof bathroom paint made by Zissner, which carries a anti-mildew warranty. It also is a wall paint, so it would not stand up to foot traffic, but it is an option in low/no-traffic spots.
Make sure you bleach-clean and then air-out and dry the area well for a week or two (or more) before doing anything. Kill the current mildew before anything, or else it will be back.