Tighten the screws that secure the rail brackets to the wall using a screwdriver. Start from one end of the stairway and make your way to the other end. Turn the screwdriver clockwise to tighten the screw in the hole, until its head falls flush with or slightly lower than the surrounding surface.
Turn a stripped screw in a counter-clockwise fashion to loosen it from the bracket. Remove this screw and replace it with an identical one. Drive the replacement screw into the hole with a screwdriver.
Replace any broken, bent or damaged bracket you come across during the repair process, since it could be the cause behind the wobbling. Position the tip of a screwdriver over each screw head that holds the damaged or broken bracket to the wall and turn it in a counter-clockwise fashion to release it. Carefully lift the bracket and tug it gently, if required, to free it from under the railing.
Hold a replacement bracket against the wall where the previous one was located. Use the screw holes as a guide to decide the location of the replacement bracket. Insert screws through the bracket holes to secure it to the wall.
Locate a new wall stud close to the previous one if the spot on the wall the bracket was attached to is severely damaged. Hold the bracket over the new stud mark and mark screw holes on the wall and the lower side of the railing. Drill pilot holes through the wall and the underside of the railing. Insert screws to secure the bracket in place. However, skip this step if the original spot on the wall is not damaged.