What is the most cost-effective way to reduce sound coming up my staircase?
The &quot;most efficient&quot; way would probably be best determined by a full acoustical analysis of your house. Since that would take time and money you probably don't want to spend...
As a way to temporarily test it out:

lay a 2x4 flat on the upstairs floor, along the bottom of the railing.
Use string to attach the panels to the 2x4 and hang them along the walls.
Move them around until you find an arrangement that seems to provide the best noise attenuation.
Permanently attach panels at these locations.

Since those panels are a bit... unattractive (unless you're into the whole &quot;recording studio&quot; vibe), consider hanging paintings, artwork and tapestries on the walls of the stairwell. Anything that has depth (paintings with chunky frames, artwork), or softness (the tapestries) will interfere with the reflection of sound waves coming up the stairs and help attenuate the sounds. A thick carpet on the stairs will help, too.
As noted in other answers, though, your best bet is going to be blocking the sounds from getting into this space in the first place. Since you have an open staircase that you can't install a permanent door on, consider a temporary door made of heavy fabric (like a blanket) that can be hung from upstairs (consider the 2x4 on the floor trick again), dropped down when you're working, and raised back up to regain the look when not needed.