Remove all of the drop-ceiling tiles from your basement ceiling.
Remove all of the drop-ceiling bars from the interior of the room. This will leave just the outside runner of the ceiling along the walls.
Use the claw end of the hammer to remove the outside runners and any nails that may be sticking down from the joists.
Check the ceiling for any drooping electrical lines, cable lines, phone lines or insulation and secure them out of the way before you start installing the drywall.
Install the drywall by securing it to the ceiling joist with the drill and drywall screws. It helps to have someone give you a hand for this part of the job, or, you can rent a dead-man lift from your local hardware store which will lift the heavy drywall for you.
Run drywall seam tape over all of the seams.
Spread the joint compound evenly over the seams and the screws with the putty knife. Keep the compound as smooth as possible.
Wait for the compound to dry and then sand it with the block sander. Once finished, apply another coat of joint compound, wait for it to dry and sand it again. Continue this until the seams are smooth and even with the drywall.
Install some crown molding around the room to cover up the seams that may still be visible where the ceiling meets the wall.
Prime and paint your new ceiling and the job is complete.