IN all the information on insulating basements I have not seen any information on the best way to insulate beside basement stairs when doing reno's.
Above the stairs we usually used 1" styrofaom and 1X3 strapping then Drywall. Comes out to 1&3/4" which just covers the stringer on the stairs usually. The spray foam it to the stringer.
Under the stinger we did the old 2x4 wall and fibreglass which is no longer recommended. Now I would do the 2" styro and then the 2x4 wall, Fiberglass insulation and no poly.
I guess you could rebuild the stairs say 2" narrower or whatever you can if they were wide enough and you still had 36" wall to wall when you were done.
There are metal Z shaped strips that you can attach to the wall then put your 2" Styrofoam in-between. Would be 2.5 inches when drywalled.
I use drill and fill method to insulate the walls along a stairway. A 2 9/6" hole is drilled in the middle of the wall in every stud bay. Then either dense packed cellulose or Retro-Foam is injected into the cavity. The hole is then plugged with a foam plug set back 1/8". 20 lb hot mix compound is applied over plug. Air sealing with spray foam is a critical step along any openings in the wall or risers. The risers are usually covered with 2" polyisocyanurate foam board with seams taped and perimeter spray foamed.