Vinyl tile installation over a concrete basement floor with moisture
One product I know of that is meant for this type of scenario is Dricore. It provides a moisture barrier from the floor.


  DRIcore has an integrate high density polyethylene membrane bonded to the underside of the panel which provides an excellent moisture barrier


I've never used it personally though it seems pretty popular in my neck of the woods.

I'd still recommend trying to rectify the root moisture problem before putting anything down.