I was wondering if it is possible to make exposed wood stairs less squeaky with a layer of foamy/rubbery material between the stringers/frame and stair treads. Or is there a different solution?
There are special screws for squeaky wood floors. You drill a small pilot hole, drive in the special screw and the head of it breaks off. The screw pulls the offending pieces together eliminating the squeak. You will want to try to screw into the stringers for the best result. With the head broken off a small hole remains that can be filled. If you have large gaps between the treads and the risers you may want to get some wood glue in the gap prior to installing the screws.
I would try the same things that fix a squeaky wooden bed; paraffin, beeswax, and/or tightening screws/adding screws.
The down side is for it to work best, you will obviously want the lube between the stringers/frame and the stair treads, so that may mean taking them off, or at least separating them some.