If you notice dampness in your basement floor, and its made of organic flooring like wood, this can lead to mold and that musty smell. There are several options to replace the affected flooring.
Some carpet tiles are composed of inorganic materials, which can provide a vapor barrier and insulation. This also protects against mold, as mold only grows on organic material.
Vinyl flooring is another great option for replacing a floor cheaply. Choose vinyl that is designed specifically for basement flooring, with the ability to act as a vapor barrier as well as provide insulation.
Laying extruded polystyrene foam directly onto your basement floor is another economical replacement solution. Polystyrene will not rot and even insulates against cold while keeping any organic materials safe from mold. Make sure to apply a sheet of plywood to the top of the foam, then firmly screw through the plywood and polystyrene directly onto the concrete using fasteners. Finally, the job can be finished with your choice of flooring.