Locate ceiling joists where the top of the partition wall can be fastened, hopefully perpendicular to the wall line or right along the top for the new wall. If the new wall location is between joists that run in the same direction, install 2-by-4 inch boards between joists to hold the top of the new wall. Put a plumb bob at the ceiling on both ends of the wall location and mark those spots on the floor.
Snap chalk lines between the plumb bob marks to locate the bottom plates for the wall. Snap one line then measure 3-1/2 inches in and snap a second line to mark both sides of the wall plate. Mark similar lines across the ceiling joists to locate the top wall plates.
Build partition walls one of two ways. Nail top and bottom plates to the joists and floor then toe-nail vertical studs to them with 16d framing nails driven at an angle with a hammer. Or you can build walls on the basement floor but cut the vertical studs short enough that a finished wall can be erected and slid into place between the floor and ceiling. Use a tape measure to determine the height or the partition from floor to ceiling and a circular saw to cut studs. Test the measurement by cutting a stud and putting it up between the floor and ceiling with a short 2-by-4 at the top and bottom of the board to mimic the top and bottom wall plates.
Make partition walls with standard framing -- studs on each end of the wall and interior studs with centers 16 inches apart. Space studs by measuring 15-1/4 inches in from a wall end to mark the location of the second stud then measure every 16 inches along the wall based on that line. Fasten studs to plates with two nails at each end of the stud, driven through the plates or diagonally through the studs into the plates.
Fasten the top plates to the ceiling joists with framing nails driven into every joist or 2-by-4 support. Secure the bottom plates with concrete nails driven into the floor in between each stud. Secure a pre-built wall by driving tapered wood shims under every stud to force the wall tight against the ceiling. Use a level to make sure it is plumb. Add 2-by-4 horizontal braces between studs halfway up the wall for extra support and to secure wallboard.
Cover both sides of the wall with drywall fastened to studs with drywall screws and a screw gun. Cut drywall panels with a utility knife or drywall saw, as needed, to fit. Cover the seams with drywall tape and drywall compound to seal the joints.