Measure the width and length of the area underneath the staircase with a measuring tape.
Transfer the measurements to a sheet of drywall using a pencil.
Place a T-square over the pencil marks. Run a utility knife over the pencil marks, cutting through the drywall. The T-square will serve as a guide to help you make clean, straight cuts.
Position the cut drywall in place under the staircase. Drill drywall screws every 6 inches through the drywall and into the vertical beams that run along the inner and outer edges and the center of the underside of the staircase.
Check that all screws heads are flush against the surface of the drywall. Drive any protruding screws into the drywall using a handheld screwdriver until the heads are flush against the drywall.
Apply drywall compound over the surface of the screws with a drywall knife. Apply the compound so that it creates a smooth, thin surface. Wipe the compound off of the drywall knife. Pull the knife over the surface of the compound to remove any excess material.
Quickly dip a piece of drywall tape into a bucket of water so that the front and back of the tape are wet.
Position the tape so that it's centered over the compound. Wet the tip of the drywall knife and place it at a 45-degree angel on the top of the tape. Place the drywall knife at the top of the tape, holding it at a 45-degree angel. Pull the knife down along the surface of the tape, securing the tape to the wall.
Allow the compound to dry for 24 hours. Check to see if the compound is completely dry -- if it's evenly white, it's dry; if it's darker in some locations, it's still wet.
Sand over the surface of the compound with a medium-grit sandpaper.