How can I repair worn edges on my carpet covered stairs?

Sounds like a tripping hazard! I would remove the carpet completely and paint the steps.

If you want to stay with carpet, how about a runner? That would cover the edges of the steps.

Once a carpet has worn thru... it will only get worse, and become a real tripping hazard..... pulling it off and replacing it or painting the stairs would probably be the best option... If your stair treads are not finished (most carpeted ones are not... where as those with carpet runners have finished treads) There are industrial grade, rubber stair treads available (and for a senior, this might be the best option)

Lots of the home improvement stores have sales on carpet and flooring this time of year, with a minimal installation charge. A minimal purchase (some purchases qualify for no interest if paid off in 90 days or 6 months) on a Lowe's or Home Depot card would make the purchase possible for a Senior with limited income (like us too) and once, paid off, the account can be closed.

Your safety is the priority here and stairs are not the place to do things half way , or to let it go any longer.... a fall or serious injury... will be more costly that re-carpeting the stairs.

How worn are the edges and what type of carpet do you gave? Is it from wall to wall of is it a runner that goes down the center of your stairs? You might call a reputable carpet dealer in your area to come out and give you ideas without charging you a lot. If you don't like their solutions just say no thank you. They should offer you a senior citizen's discount as well.

Actually you can remove the carpet and paint the stairs. Put on a coat or two of polyurethane on top of the paint. Maybe brown steps and white risers? It would look much better and be long lasting without costing very much. It also would be safer than having worn carpeting.