Styrofoam has almost ZERO sound deadening properties, and about the same absorption ability. For sound PROOFing, you want lots of MASS in only two leaves; each leaf can have more than one ply, or layer - for example, 2 plies of wallboard, separated by studs, then 2 more plies of wallboard.
What you want to avoid is having more than one air space in a sound wall, or more than two leaves.
Before you do anything else under the stairwell, you should turn all the lights on OUTSIDE the stairwell then go inside and close the door (no light inside) - if you can see ANY source of light, the basic construction needs to be tightened up.
Both sides of your framed in area should be covered with at least one, and preferably two, layers of sheet rock - each layer should be caulked before the next layer goes on, and seams should NEVER be lined up with each other. EVERYTHING you add should be caulked with at least a high quality silicone caulk, and it would be much better if you could find some actual acoustical caulk, which is difficult to find locally.
The end result of your building should be a box that is hermetically sealed when you close the door - any little crack will let sound in/out, and lower the isolation you can achieve.
As to sound TREATMENT, which has nothing to do with sound PROOFing, if you can't find a local source for rigid fiberglas board, such as Owens Corning 700 series, or Knauf board, there is a more expensive route of using the so-called acoustic foam on the inside surfaces. None of the cheaper attempts can do the job - trust me, it's all been tried and the best attempts have been marginal at best... Steve
