A handrail is only as useful as the wall behind it. Most of the work is finding solid anchor points, marking them correctly, and proving the rail can take a load before anyone relies on it. Follow the steps in order, and do not skip the test at the end.
Local option: New Plan Care gives families in the St. Louis area in-home help so a parent can stay home: companionship, help with bathing and dressing, meals, and overnight care. Call (314) 405-0887.
Before you buy anything, read the instructions for the rail
Every rail system states the bracket spacing, the fastener type, the rail height, and the load it is designed to take. Write those four things down before you go to the hardware store. If the instructions say the rail must be mounted into a wood stud or a specific anchor, buy that fastener, not the one that looks close.
Find the studs and confirm them
Use a stud finder along the wall where the brackets will go, and mark each stud edge. Confirm each one with a small test hole in a spot the bracket will cover, so the hole is hidden after install. If the wall sounds hollow, crumbles when you drill, or is plaster over lath with no clear stud, stop here. That is a job for a contractor, not a weekend.
Mark the bracket line level with the stair
Use a level to set the bracket line parallel to the stair slope. Mark the height from the step nosing as the instructions specify. Check the marks against each other and against the stud locations. A bracket that is off by a little on the wall will show up as a rail that looks crooked for years.
Pre-drill and mount the brackets
Pre-drill each anchor point to the depth and size the fastener requires. Drive the fasteners straight, and tighten them until the bracket is firm without stripping the hole. Do not overtighten into drywall or soft wood, because that is how a bracket pulls out under load.
Attach the rail and check the fit
Fit the rail to the brackets according to the manufacturer's sequence. Tighten the set screws or caps as directed. Run your hand along the full length to find any gap, rough edge, or spot where the rail sits off its bracket.
Pull-test the whole thing
With a helper watching, grip the rail at several points and pull firmly outward, downward, and along the rail. Watch each bracket for movement, flex in the wall, or any sound of a fastener shifting. Repeat after a day, since a fastener can settle. If any bracket moves at all, take it off and fix the anchor before the rail is used.
Keep a check on the schedule
Write down the date of the install and check the brackets and fasteners on a set schedule you choose. Recheck sooner after any bump, heavy use, or a visit from a helper who leaned on it. For support in finding local help with home projects, see Local Help for Caregivers in St. Louis.