I have a stone path that is all broken up. What's a quick and cheap way to fix it? Please see the picture. There are several areas in the path that have this issue. Would putting a strip of concrete 1" wide all along the path would help keep them in place?
It seems that your problem is likely due to movement of the supporting soil under the stones, so your idea of a bordering strip to stabilize lateral movement is a good one. A strip of concrete only an inch wide would be difficult to create and probably ineffective. You would be better off with a larger border.
The method you choose to create a border would depend on how you want it to look, how much work you are willing to put into it, and what materials you are comfortable working with. Concrete would work if formed and poured properly. I would probably use a quick and easy method, like pounding rebar in to support partially buried pressure-treated dimensional lumber (1x4, 1x6, 2x6, etc.)
You will need to pull up the stones in the whole area, create your borders, than backfill with sand and tamp well until the area is solid and level. Then replace the stones and fill the joints with sand and tamp again.
Kinda like the picture below, except with lumber instead of flex border: