For a dry, mold-free finished basement floor, always install a vapor barrier before laying a wood subfloor or carpet pad.
Any kind of persistent moisture will allow mold to get a foothold and soon ruin carpet or wood flooring. For below-grade slabs, assume that the concrete floor will get damp at some point. You then have two options, depending on your circumstances. And both of the options use the same layers of 1/2-in. plywood, carpet pad and carpet as shown. It's the initial layer that differs.
Option B (in the photo) will work on a concrete floor that has no persistent dampness, seepage or leakage. The 6-mil layer of plastic helps to minimize potential moisture migration up into the plywood. Option A can be applied on concrete where there's a higher risk of some dampness. The initial layer is a durable high-density polyethylene sheet or similar system that uses evenly spaced 3/8-in. tall dimples to create air space and a moisture barrier between the concrete and the plywood.
Lay the sheet over the concrete floor (dimples down), overlap adjacent edges and tape the seams. Add the plywood layer on top and anchor it to the concrete with 15 concrete screws (predrilled and countersunk) per 4 x 8-ft. sheet.
Dimpled polyethylene underlayment is available from home centers, flooring suppliers or online.
Before you proceed, consult a local building inspector to determine specific building codes for this type of project. Also, be sure your basement floor is level. Finally, note that these options will raise your floor by 1-1/4 to 1-1/2 in., so make sure this added height won't create problems.