You need to be careful when removing staircase sidewalls. If the stairs aren't supported properly, they could collapse. Do you need to remove both sides and if the answer is no then what side is it that you want to remove?
Most stairway walls can be removed because the stringers are tied in at the top and they rest on the floor in the basement. There may be a post at the bottom so as to tie a handrail to. Leave one wall up for a handrail.
If you have a Sheetrock wall then chances are there is nothing there but non structural 2x4's. Start by removing the Sheetrock and see if it's a non-bearing wall. (I don't know your house so be sure it isn't load bearing). The way to tell is to see if the wall is carrying the roof line unless you have wood trusses up in the attic. If the peak of the roof runs parallel to the wall you want to remove, it is considered a bearing wall unless, like I said, you have roof trusses that sit only on the exterior walls.
You can then remove all 2x4s (if they are not load bearing). and this will open your sidewall. If you want to support the floor above the wall where the 2x4's were removed all you need to do is carry a beam (header) from one end then run it as far as you feel you can go without losing the openable area you are looking for. You may leave four feet from the bottom stair inward towards the top step along the floor above.
You would then need a support collum. A steel adjustable colum or a 4x4 would do. This is probably the cheapest way around your problem. If you have a basement then, unless it is a raised ranch, you will need to dig down at the exterior grade level to match an area, (since it is a bedroom and subject to National Codes) where the window bottom sill is no higher than 44" off the floor inside the house.
This tends to create a water and moisture problem because the exterior grade is probably around 54" so you would need to dig down 10 inches below the exterior grade level. (I am using typical house measurements so your particular home may not have the exact measurements I am using).
If the house was built to code 9 years ago and it has a Certificate of Occupancy then you know it was pretty much built according to the National 1 and 2 family building code that all building inspectors use nationwide. This is what I am assuming your house was built by when I am attempting to measure up your foundation depth, exterior grade level minimum clearance of 8" and ceiling height. One more thing, bedrooms are not allowed in basements anywhere in this country so, (although some towns are more lenient than others on this) if you must use the room for a bedroom you will need a window that is no less than 32" in height by 30" wide with a bottom window sill of no less than 44" off the basement floor so a firefighter can go in and get you out.
Hoped this helped:)