You can install laminate, engineered wood, resilient flooring, and many other finished flooring materials directly over a dry concrete slab. However a wooden subfloor helps insulate the flooring from the cold concrete. It also adds a little cushion to the hard surface of the concrete. A wooden subfloor will not cure moisture problems. Moisture that comes up through the slab will ultimately find its way into your finished floor. Cure the moisture problem before you install the floor.
Headroom, especially in a basement, may be at a premium. To make low flooring framing that will conserve headroom, lay pressure-treated 2x4s (sleepers) flat and cover them with 3/4-inch tongue-and-groove OSB underlayment or exterior plywood.