Cut two 5-inch lengths of 2-by-4-inch board and two 12-inch lengths of 2-by-4-inch board. Place these around the perimeter of the pole, broad side down, forming a square. Nail the boards into the subfloor when they're positioned.
Measure from the top of the base boards -- those you've just installed -- to the ceiling joists above the base. Cut four 2-by-4-inch boards to match the distance.
Place these boards vertically onto the base boards and wedge them underneath the ceiling joists, forming a square around the pole. Toenail the boards into the base and the ceiling joists with two 8d nails at the top and bottom.
Cut four pieces of drywall with a utility knife, one each matching the measurements for each side of the framework on the post. Nail each piece onto the frame with drywall nails, spaced every 6 to 8 inches.
Nail drywall corner beads onto the corners of the post covering. Apply joint compound to the corners and let it dry.
Sand the joint compound until it's smooth and even. Prime and paint the frame to match your basement decor.