stairstrim
We recently added wooden flooring to our stairs. I'm looking for a trim to add between the stairs and the wall. Something that I can cut easily and glue and then caulk. I'll need to edge it (rounded?) for the lip of the stairs and angle the inside to join up with the vertical. Is there anything made for this application?
How do you plan on not painting if the new trim is lower than the old trim? Personally, I'd go with the same if not larger trim to hide any imperfections left by the old trim, but maybe that's just me.
To remove old caulk, just use the flat edge of the painters multi-tool to scrape it away. Be careful not to dig into the wall or pull up the paper from the drywall. For anything left, sand it down. You'll probably want to have some extra drywall mud or spackle available to fix any spots and to cover any screw/nails that were previously covered by the old trim.
And then, yes, you'll need to paint. It's easier to do a first coat on the trim and the walls before installing the trim. Then all you have to do is touch-up any nail holes or scrapes.
I'd get a 2 x 4, trim it to fit the length and height of your trim piece and screw it in. 
The horizontal lines are the piece of trim that will run along the outside of the staircase.
Best Answer
I personally think adding trim will just draw more attention to the side of the stairs. And no matter now nicely you trim it, it will stand out as people are not used to seeing trim on stairs.
In my opinion, I would just use caulk to fill the gap between the existing skirt and treads. The gap looks pretty small and most people will never realize what you've done.
Edit: I just realized... There is no skirt board in those pictures. You can add a skirt board after the fact but it is a very time consuming task. See the link for an idea of how to do it. http://www.finehomebuilding.com/how-to/tips/scribing-a-skirtboard-to-an-existing-stair.aspx