No offense intended, but if you are asking this question, you should probably plan to strip and refinish the whole thing. There are techniques for adding in color, but staining a previously finished wood surface is not one of them.

Do not be tempted to try Polyshades.

What would be the best product to use to strip? I have to be careful as the bannister rail is stained but the spindles have been painted and I do not need to refinish those.
Thanks

Below is an article on stripping. If you are in the temperate northern hemisphere, you will probably want to wait for days where you can have the windows open to facilitate adequate ventilation.
About strippers
About stripping

Someone familiar with shellac and could very easily apply more shellac in just the right 'tint' to get your railing back up to a good appearance. Once that is done, then a careful topcoating with a quality polyurethane would give you a soil and oil resistant protective finish.

Stripping is not always the best way to go.

That was sort of my inference. It can be done, but if someone asks the question above, they probably are not experienced enough or have access to the resources to do it.

In addition, if the shellac is not de-waxed, polyurethane may not adhere well.

"Someone familiar with shellac and could very easily apply more shellac in just the right 'tint' to get your railing back up to a good appearance. Once that is done, then a careful topcoating with a quality polyurethane would give you a soil and oil resistant protective finish."

Very true...the shellac needs to be a de-waxed product. We use de-waxed shellac prior to coating wood flooring with polyurethane finish.

As for strippers...NMP may not be a cancer-causer, but it is something to work very carefully with.

Here is a link that might be useful: N-Methylprrolidone (NMP)

Our bannister is quite old & has a shiny topcoat over stain. I want to re-stain it & cover with either shellac or polyurethane. Which is best? Must I sand down to bare wood or simply remove the protective topcoat?

Read this article before you start. Polyurethane does not stick well to old finishes, so I would rule it out.

Stains are meant to be applied to raw wood (or lightly sealed wood). Glazes are meant to apply between coats of wood.

Before you go thinking of re-staining, clean it and wipe on some mineral spirits to get a preview of what it will look like with a new top coat. You may not need to add color.

Here is a link that might be useful: Saving the finish

Handrails are an especially difficult project to get good adhesion on. After refinishing literally thousands of feet of them, I have come to expect to find the wood will be coated with a soft, opaque slick muck of years worth of hand oils worked into the pores of the wood. Dewaxed shellac is my favorite finish to put the first color coats on but nothing, I mean NOTHING, will stick to that hand oil muck. I would FIRST use a 1 1/2 inch good quality natural bristle brush ( for best brush control with harsh solvents, less to fix elsewhere later) to apply a thin layer of methylene chloride stripper, with plenty of plastic and old towels around to stop or catch drips. then clean and rinse with the same brush applying denatured alcohol, 3m green scrubby wool pads, wipe down and dry with plenty of disposable rags or blue shop paper towels. sand to 220, tack away dust. tint dewaxed shellac with analine dyes like homesteads TRANSTINT drops to taste. Shellac alone will not withstand years of wear so I topcoat with a good quality varnish. my preference is behlens rockhard tabletop varnish but most poly or varnish will do. buff the final handrail by wetsanding with 400 or 600 grit paper with paint thinner or orange oil as lubricant, or wax with 0000 steel wool.. handrails need to feel sexy smooth. don't skip the final polishing or your hand will find every speck of dust chunky every time you go downstairs... ugh
good luck and enjoy!
Reid