This is the Hungarian Embroidery bookmark that got Deborah to ask her question. The edges were folded over with enough area folded on the back to be caught by the herringbone stitches on the front. The double lines of stitching on the back is the back side of the herringbone. For extra security you could use your sewing machine to do a small zig-zag stitch on the long sides before folding them over and doing the herringbone. I did this with the small one (could not get a good picture since the sewing thread matched the bookmark and my stitch was small). I did have to remove some of the zig-zag at the top and bottom areas that are fringed in order to get it to fringe though! I suppose you could also do it by hand similar to what you would do to keep your fabric from ravelling while stitching on it, but I always use my machine. Also perhaps surging the sides before folding over. I would do this after the stitching is done and before the herringbone to make sure you leave enough fabric to catch on the back with the herringbone.
This is another Swedish weaving design. for this one I used my sewing machine and did small straight stitches around the four sides and then cut the fabric outside the straight stitches to the length I wanted the fringe and then fringed it.
These are all the bookmarks I could find at my house and I think it covers all the ways I have finished them. There could be other ways, but hope this helps some of you or gives you some ideas. Please share with the rest of us, if you want!