The stairs in our home being built (production builder) is concerning. The gaps in risers and treads doesn't seem right. When seeing all other homes in same neighborhood under construction they do not have gaps and risers are plywood allowing a butt to be flush. There is carpet going to be installed so the builder is not concerned about gaps. I know it will be covered but years later what problem will this cause when we want hard finished stairs. My other concern is there was no wood glue used at all. Will I be dealing with squeaky stairs in a year? What things can I do myself to ensure no squeaks? Drywall has already been installed so I have no access to underside. 
New home with gaps in stairs
stairs
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There are 2 ways to do this, the first way is in essence just cutting the wall at the top of the skirt board and capping it off and setting the handrail on top of the cap that is finishing off the cut wall, sounds a bit crude, but essentially that describes what you will have.
The up side to this is you will not need to do a floor repair, depending on where your wall ends where the newel will be.
The other way to do it is by removing the wall entirely, repairing the floor, removing the treads and risers, and set longer ones with return nosings on either side. and re-support the stairs with a new walls under the existing carriages.
This is the most work, but will give the best result in my opinion.
Jan. 2, 2014 edit
Here is the work I did in 1989 that shows what I did.
This one shows the best example of the effect I was trying to explain. I do not have a shot that shows how the handrail goes past the ceiling, but with a little imagination you could picture what the rail would have to do if the stairs were much wider at the bottom.
Edit #2
I did a little calculating to help, if it may
If the cleats supporting the steps are solid and haven't deteriorated, use 3 inch course-thread screws to re-secure them to the stair stringers (the long outer sections that span from the bottom to the landing.
If you are in doubt as to the condition of the cleats replace them. I'd use any 2 x stock cut a bit longer than the width of the step. Install it so the cleat's edge is recessed back from the step's edge. From the looks of your foggy photo it appears your steps are 2 x ? stock. If they are in need of replacement (and can't be repaired) replace with similar dimensioned wood.
Also you might want to consider if there are several steps and/or cleats to be replaced is to install a third stringer between the existing ones. They can be found sometimes pre cut. And it isn't that hard to cut one from a length of 2 x 10 or 12 Douglas fir.
Best Answer
Those look like temporary steps to me. For carpeting you usually see 1-1/8" bullnose particle board treads and 1x8 or 3/4" plywood risers.
Rarely is two-by lumber left as a final product, as it's heavier-duty than necessary and creates very thick and squarish nosings when carpeted. We'd commonly do something like that, or doubled scrap plywood, until after taping and painting is done, then install the final materials using construction adhesive.
Yes, it should be done better, but that's my opinion. It really comes down to your relationship with the builder and your contract. Ask your builder if that's the final state.