My deck stairs which are supported by ledger strips keep pulling away the stringer. Last year I used a clamp to squeeze them back together but they pulled away again. The railing also is wobbly. What would be the best way to fix the stairs so they are more stable?
Hard to tell whether you have screws or nails holding everything together right now. It is either nails - which can, relatively easily, come apart, or screws which can come apart but not as easily.
If you have nails then I would reinforce using some long (e.g., 3") screws and you should be set. I would want two screws into the end of each tread. The railing is sometimes a bit trickier, but screws to supplement the existing nails is a good start. If that doesn't take care of all the wobbling of the railing then you may need to replace parts or add some additional supports. Of course, if you find any cracked or rotted pieces, or if anything cracks when you drive in screws, then you need to replace those pieces.
If you already have screws then I'd be a bit more concerned as that may be a sign of the wood rotting (and therefore not holding the screws any more) or that there is just too much movement of the ground causing everything to come apart.
Screws are a good idea.
Another fix that also would help vs flex would be to put a 2x4 under each step, oriented so the 2 inch side is up against the bottom of the step. Attach the 2x4 to the step and the ends of the 2x4 to your stringers.
I would scout the neighborhood for scrap 2x4s for that use. If you are going to buy new wood, replace that splintery looking top railing with the nice new 2x4s and use that old railing wood for under the steps.