Every staircase in and around your home should have stair handrails on at least one side for safety when using the steps. In some areas, it is required by city code to have handrails, sometimes on both sides of the stairs. Anything used as often as a handrail is can be subject to dirt, stains, scratches and other damage.
The two best ways to protect the handrail from scratches is to paint it with a gloss enamel, preferably an oil-based product for durability. If the handrail is stained, apply a couple of coats of good varnish to protect the wood for many years.