I had an old Victorian with beautiful woodwork like that, and I used a fine steel wool with a mild oil soap and just scrubbed it well and then wiped it dry. The dark did indeed come off and it all lightened up nicely to what I imagine was the original finish. It took a little elbow grease, but I think I only spent a day on it and there was LOTS of woodwork.

What you don't want to do, unless you want to get into an whole refinishing project, is to remove the finish entirely. You may want to try a spot with the steel wool and soapy water to see how that does.

can you tell what kind of finish you have?

When we moved into our 120 yr old house I gave my daughter a rag w/murphy's oil soap and told her to go at the staircase. I came back a half hour later and she proudly showed me where she had "polished" a newel post. She sure did. :)

Carpet it and move on to the next problem.

kterlep -- I am completely unversed in all of this, so I have no idea what to look for in determining the finish. I'd be happy to figure it out with some pointers, though!

mightyanvil -- Carpet is the last ditch option. But, yeah, it's still an option. We do want to put a runner down, but I am thinking it won't cover enough of the icky parts, right?

I agree with using fine grade steel wool and a very mild soap. The paint and dirt will come up. I wouldn't use anything else till you tried the steel wool and mild soap.

Oh my goodness, even with the "dirt", you still have beautiful stairs and woodwork. Please, don't paint. I had to do the same thing in my great grandmother's house a very long time ago. I still remember that. I was in my early teens and I was quite proud of my results. Just a good memory. They really should clean up well with some mild soap and water. Steel wool would be fine and just a really good scrubbing. I would love to see your results.

The good part about the stairs being "dirty" (I think you should say "patina'd") is that the paint will come off easier.

Don't do this on the top of a step, find an inconspicuous spot. :) Here's some ideas for figuring out what your finish is:
turpentine - if it dissolves, it's wax.
denatured alcohol or Lacquer thinner - if it dissolves, it's shellac
xylol - water-based
paint remover - Polyurethane or varnish

I'd wait for someone brighter than me to chime in. :)

You find that kind of dark shadowing even on newer furniture. We have an oak living room set, Amish made, and on one arm chair, the dark has started where body oils have accumulated on the wood. It's almost 'gunky' and you feel like you could scrape it off carefully. Oils oxidize and darken, just like zits. LOL.

Back when I was a child, wooden floors were the norm and not not the exception and wooden floors and furniture was oiled to keep it looking shiny. It was common for the lesser used part of a floor or stair tred to darken from lack of wear.

I think your project may turn out to be less of a problem than you think, and the wood looks like it's in good repair. The steel wool and soap will remove paint spatterings. I use it routinely on my wood floors when I find paint spatters, and that's so common when people use rollers, as opposed to brushes.

I have used Howard's Restore-a-Finish with some success.
I have also used mineral spirits and quadruple O steel wool. HTH

Beautiful stairs! I would figure out where your runner is going to be and test on areas that will be covered. Conversely, once you figure out the best method for clean up, put most of your elbow work into those areas that will show.
Try to avoid painting, they probably still won't look very good, and you'll make it nearly impossible to ever get them restored.