Measure across a sheet of drywall and place a mark every 16 inches. Use these marks as a guide for the ceiling joists' positions when the drywall is raised overhead. If the drywall sheet will cover lights, measure from the wall running perpendicular to the drywall to the light fixture's center and from the wall running parallel to the drywall to the light fixture's center. Transfer these measurements to the drywall to indicate the light fixture's center. Repeat for all light fixtures the drywall overlaps.
Lift the drywall sheet up and place it face down on a drywall lift's arms. Crank the drywall lift up to lift the drywall up to the ceiling joists. Roll the drywall lift so the drywall's corner is positioned in the corner of the basement ceiling.
Secure the drywall to the basement ceiling joists using drywall screws. Use one screw every 6 inches and sink the heads slightly below the drywall's surface. Crank the lift down and move it out of the way.
Cut holes for the light fixtures out of the drywall with a drywall cutting tool. Poke the tool's cutting bit through the drywall on one of the marks you made in Step 1. Move the cutting tool's bit to one side until it hits the fixture's side. Follow the resistance you feel from the fixture to cut out the hole. Repeat with the remaining fixture markings.
Measure the gap from the drywall's edge to the opposite end of the basement's ceiling. Transfer this measurement to a new piece of drywall. Hold a level against the drywall at the measurement mark. Cut the drywall along the level's edge with a utility knife. Hit the back of the drywall, behind the score line, to break the drywall. Cut the drywall's paper backing.
Mark any ceiling light fixture's placements onto the drywall, the same way you did for the first piece. Position the drywall on the lift and crank it up to the basement ceiling. Secure the drywall to the basement ceiling joists with drywall screws and cut out the light fixtures' holes. Continue positioning drywall on the basement ceiling until you have covered the entire ceiling.