Maybe its the photo but the stringer on the right looks like a 2"x2"?
I like the idea of a centre stringer.
A better look at the stringer. Stringer doesnt even appear to tie into anything. Its just out there.
Hold the drywall, we have a concern! Something does not look right with the way the open web trusses rest at stairs. Should be solid blocking? How are the stairs attached? I am not crazy about the plywood for stairs, it certainly looks low end construction. I guess the 2x2 is holding the stringers in place …Ooops. 
Follow on question. Is there any requirement that anyone knows of that would require a door accessing a closet under a stairwell to be fire rated? The area under the stairs is completely finished out with 5/8 drywall. Another inspector in town says the door accessing the closet needs to have a better casement. It is a standard hollow door with a wood casement. No threshold. I think it could be a louvered door and still pass, as long as the area under the stairs are enclosed properly.
Ask the building inspector to put that in writing and to provide the building code requirement so that you can do the job properly. In other words call his bluff.
1.Yes it should have fire blocking if there is a ceiling on the other side. 2003 IRC R602.8. Why? Because if there is a ceiling on the other side then each bay is essentially a flu if the blocking is not in place.This flu effect will cause flames to spread rapidly.
2.If you gain access to the understairs by means of a door.then it must be sheetrocked w/at least 1/2" dry wall. 2003 IRC R311.2.2
The only time I’ve ever had to put a fire door in the basement was to separate living space from utility area.A building inspector said it was required but I never did find it in the IRC.Maybe it was a local thing.It was easier to do it than to argue it.Besides it made sense to me.
Cheremie
That doesn’t look like a stringer to me. My guess is that the carpenter just used a skirt board (the light blue board) and routered spaces for the treads and risers to fit into it. Then the carpenter fastened a 2 x 2 underneath for some additional support. The thickness of the 2 x 2 appears to stick out past the skirt board, so what is above the 2 x 2 doesn’t appear to be the standard thickness of a stringer. Both the skirt board and the 2 x 2’s are fastened into the wall framing for support. If that’s accurate then this stairwell installation is missing both of the side stringers along with the middle stringer and the stairwell header as well as the fireblocking on the side (the stairwell header and plywood header board, both missing, would have provided the fireblocking where the stairs should have met the stairwell header). Once the framing is corrected, adding drywall, and taping all seams and corners will provide the fire-rating.
I concur with your reading of IRC R311.2.2. The walls, as well as the “ceiling” under the stairs, and a believe a small section of horzontal ceiling are I believe 5/8 in drywall. But they certainly meet R311.2.2. I am having a harder time visualizing how R602.8 comes in to play. I think it is met by R602.8.3 which refers back to R311.2.2, because it is a door leading to an area under the stairs. It doesn’t say the door needs drywall. Now the access door to the crawlspace, that one has drywall on it.
If building inspectors are going to make demands that supposedly required under the building code they had better be able to quote the chapter and verse in writing. Thats their job, and they have duty to do so. They can only recommend whats in the code book or local requirements.
Adding a door in this situation, a threshold is nonsense in my opinion if the understairs are drywalled. As to the framing well big deal.
This photo must have been taken in La-la-land, where everyone is beautiful, where anyone with nailbags and a twenty-two ounce hammer is a carpenter, where municipal inspectors are endemnified against any responsibility, where private inspectors disclaim code-compliance inspections but get sued anyway for code defects, so that insurance companies can roll-over, so that attorneys and clients can make outrageous amounts of money, so that everyone can go on living carefree lives “in the warm California sun.”
Actually Texas!
If under stair protection is provided to provide the same fire protection as everywhere else in the house ( 1 hour), would that not be sufficient and not have to provide a threshold and fire door??? What would be the benefit.???
Marcel 
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The benefit? There is no benefit. It sounds like code inspectors not knowing whats in the book they think they are quoting from. To be suggesting a fire door on a lined closet is irrelevent in the stopping of fire spreading.
Stringer is nailed to wall studs.
[quote=rwand1]
It looks like new construction. And the schematic above shows that two joists should be used as a header, they also act as fire blocking.quote]
The schematic shows framing for a framed stairwell opening in an unsupported floor. In the photo, the joists are bearing on a wall, so structurally the double joist head-out is not neccessary to support floor framing. Stringers are nailed to wall studs so the staircase is adequately supported.
The main concern in the photo is firestopping. I see this condition (no drywall on bttm of staircase) all the time in basement staircases. Firestop rules have changed over the years and the requirements are easier to understand in some situations than in others.
Either way, it’s a code issue. If the area beneath the staircase were enclosed and contained mechanical equipment which could start a fire I’d call it out as a safety issue because a fire could move into the joist bay in which the block is missing. Or if the fire could burn through the staircase and into a multi-story stairwell, thereby spreading fire quickly throughout the home, I’d call it as a safety issue.
Kinda depends on the situation to me.
The framer should have blocked over that wall, so… it’s a missing block at least.