A handrail should feel solid before anyone relies on it. A loose bracket may only need a fastener tightened, but movement can also mean the anchor has pulled from the wall or the bracket was never attached to a suitable structural point.
Check the movement
Hold the rail near each bracket and apply gentle pressure. Identify whether the rail, bracket, screw, or wall is moving. Do not use a loose rail for support while testing it, and keep the stairway clear of tools and people.
Inspect the fastener
Remove the decorative cover if the design allows it and check the screw head and mounting plate. Tighten gradually with the correct driver. Stop if the screw spins, the plate bends, or the wall surface crumbles.
Repair the anchor
A bracket that has pulled out needs a suitable structural anchor or a new location aligned with framing. Do not rely on a small drywall plug for a handrail that may carry body weight. Avoid drilling until you have considered hidden wiring and plumbing.
Test the finished rail
After the repair, check every bracket and apply moderate pressure along the rail without jerking it. If the rail remains loose, the wood is cracked, or the stair geometry is unusual, use a qualified carpenter or building professional.