Cut a 12-by-12-inch square out of the drywall ceiling below the staircase using a drywall saw. Take a peak inside with a flashlight to see what the frame inside looks like. Cut the drywall along the frame with a drywall saw and remove it.
Run a stud finder along the wall underneath the staircase. Mark the location of each wall stud you find.
Drill a pilot hole through the staircase's skirt board below the stairs and into each wall stud. Use a drill bit slightly smaller than the diameter of the lag screws you will be using. Attach a nut driver to your drill and drive one lag screw through each pilot hole to pull the stair treads tight to the wall.
Place a tube of silicone caulk into a caulk gun and remove the tip with a utility knife. Run a bead of caulk along the seam where the skirt board meets the wall.