A loose handrail bracket can make a staircase less secure. The repair should restore solid attachment to framing or an appropriate wall anchor rather than simply tightening a screw in a stripped hole.
Support the rail
Have someone steady the rail or support it temporarily. Inspect the bracket, screws, wall surface, and rail for cracks.
Find solid structure
Where possible, align the bracket with a stud. If the wall is hollow, use a load-rated anchor designed for the wall material and handrail load.
Replace damaged hardware
Use screws suited to the bracket and substrate. Do not reuse stripped screws or a deformed bracket.
Test gradually
Tighten evenly, then apply light pressure to the rail at several points. Stop if the wall flexes, cracks, or the bracket pulls away.