Insufficent Joist Support

I am going to attempt to paint a picture first, if this does not work then I will upload a picture. The picture I am painting is what I am guessing happened.
We have an older home...1950's. From what we can tell, there were a few mods done to the house over time.
A normal kitchen...about 12 x14, with steps leading down to the basement. Well...now picture that when the basement access was cut out in the floor, that left several floor joist shorter than they were when they extended to the block wall on the other side of basement.

My question is, it seems no matter how you try to take this load to the load bearing block wall in basement....you would have to have support here ( where the floor joist are cut)?

I do not see how framing techniques could compensate for this.
Anyhow, where I have "no support here", there is a wall directly over this and thus this floor runs down hill to this area. The "no support here" includes the entire line. It is not a main load bearing wall but I would like to lift this back up(its over basement)and add some support to keep it from sagging again. Any suggestions would be appreciated.

You can use jack posts and jack it up, then build a supporting wall to support it and then you can take the jack posts out.

Bryce,
I appreciate the suggestion. I am not against a framed wall but do have a few other questions. Is there anyway I could support it in 3 places with blocks? The basement is already small so I would prefer to keep it open as much as possible. Second question, if I do use a framed wall, is there any special considerations other than pressure treated on bottom plate...would I need to use something other than 2x4's or would I need to have them on shorter centers? Thanks for the feedback.

"Is there anyway I could support it in 3 places with blocks?"
- yes you could build a concrete wall, is your basement wet? then yes

" if I do use a framed wall, is there any special considerations other than pressure treated on bottom plate...would I need to use something other than 2x4's or would I need to have them on shorter centers? "

You shouldn't use treated wood indoors-- VOC's, better to cover with plastic 6mm or so between the concrete and wood. I would try to make it strong as possible you can google some framing techniques, but 16" is the usual. Remember to use a level...

You can buy some jack posts for about $60 each, that would be the fastest.

Your floor joists appear to be in mid-air according to your drawing. I am assuming there is no support underneath them.

You do not have to build a full wall to support this structure. You can rent some screw jacks to temporarily re-level the structure (cheaper than buying). Since the span looks to be relatively short, you can install a 2x8 header (sandwiched 1/2" ply with two 2x8s) to distrubute the load. Then, you can install wooden posts (probably 4x4 seems sufficient) at each end to carry the load to the floor. The pure builder would want you to cut into the basement floor and pour a proper footing. I leave that decision to you. Once the new structure is complete, remove the jacks ...

This approach leaves your basment more open than a full wall or block wall. Give it some thought.

The full length joists at end of thew stairs should be two joists nailed together and the rim joist that you have said is not supported should be two joists nailed together. In a new house some of this would likely be engineered beams. The best way to fix this would be to figure out just what is sagging and change the guilty part for a beam or two.

Check out page 50 here and notice double joists and joist hangers.
http://www.engr.sjsu.edu/dmerrick/164/framing/ah73chapter3.pdf