Stair stringer sizes are famously under-prescribed in the International Residential Code (IRC). There's no way for a DIYer to determine if that wall is structurally necessary (nor for a home-builder, not that this stops them). If pressed, I would probably hack the IRC's floor joist tables.
The following figure demonstrates the math process converting a stair stringer's live load of 40 psf to an equivalent live load applied on a horizontally oriented floor joist, where I simplify things by isolating a section of a stringer that would cast a 1 ft shadow at high noon:

First I get the vertical force by multiplying the 40 psf by the shadow width, 1 ft, and by 1 ft of loading width perpendicular to the screen to get a 40# force. This force isn't perpendicular like a floor joist's live load should be, so I break the loading down into an axial component along the stringer's long axis (which I'll ignore as inconsequential) and a component acting perpendicular to that long axis. Taking the stair slope as 11" run by 7-3/4" rise, I reduce the 40# to its perpendicular component with (40#)(cos(arctan 7.75/11)) = 33#.
Now I want to convert this back to psf, so I'm going to divide it by that same 1 ft perpendicular to the screen, but for the other dimension I want to use my isolated section's length along the stringer's long axis: 33# / [(1ft)(1ft/cos(arctan 7.75/11))] = 27 psf.
The IRC contains a joist table for 30 psf live loads (for bedrooms). Since the 27 psf live load computed above is less than this 30 psf, but still somewhat close, the 30 psf live load table will provide conservative stair stringer design lengths that aren't terribly overbuilt. Measure from a stair notch down to the bottom edge of the stair stringer (the shortest distance, not straight down) and round down to the nearest dimensional lumber size, e.g. 6-1/4" ↦ 5.5" implies 2X6. Now go to the 30 psf live load + 10 psf dead load floor joist table and look up 2X6 (for the 6-1/4" demonstration case, not necessarily your case) for your grade and species of stair stringer lumber. The table will give you a conservative maximum length (measured along the stringer's long axis) for your stair stringer.