A stair rail is judged by one thing before anything else: whether each bracket is fastened into framing that will take a grown man pulling on it with his full weight. Drywall anchors, hollow-wall toggles, and screws that only reach the plaster will look fine on the day they go up, then pull out the first time someone grabs the rail on the way down. Decide the mounting question first, then decide who does the work.
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Check the wall before you check the price
Find the framing where each bracket will go. A stud finder is a starting point, not proof, so confirm by drilling a small pilot hole where the finder says there is wood, then checking the depth and the feel of the bit. If the wall is plaster on lath, masonry, or a stair stringer you cannot reach, the job changes. Write down what you find at each bracket location so the installer can read it without guessing.
When doing it yourself makes sense
A straight run of rail into solid wood framing in a drywall wall is a reasonable DIY job if you have a drill, bits long enough for the screws, a level, and fasteners rated for the job. Bring a second adult. After installation, put your full weight on the rail at each bracket, then have a helper pull sideways while you hold. Any movement at a bracket means the fastener is not in framing. Stop and fix it before anyone uses the stairs.
When to hire it out
Hire the job if the wall is plaster or masonry, if the rail needs a return or a post at the top or bottom that has to be fixed to a newel or framing you cannot verify, if the stair is open and the rail has to fasten to a stringer you cannot inspect, or if you cannot find solid framing where the brackets must go. A loose return or post is the failure that costs the most, and fixing it usually means opening the wall anyway.
What to ask the installer
Ask which framing the brackets will hit and how they will prove it. Ask whether they will do a pull test with a person on the rail before they leave. Ask what fasteners they will use and whether those fasteners are rated for the wall they are going into. Ask whether they will confirm the rail height and grip size against the local building department's rules, and get that answer in writing. Compare quotes on the written description, not the hourly rate alone, because a cheap install that moves at the bracket is the most expensive rail you can buy.
Dad's side of it
He may not want a crew in the house. Tell him the plain version: the rail goes in during one visit, he can be there for the pull test, and he gets a say in where the rail sits. A man who checks the rail himself is less likely to resent it.
The next move
Tonight, put your full weight on the existing rail or on the wall where a new one will go, and note any movement you feel. If you want to see what this desk relies on before you decide, the Sources page is the place to start. Then call one installer and ask the four questions above before you ask for a price.