Not sure without seeing the quilt, but I think this is what I would do. When you come to an unravelling place, stop and remove some of the stitches. Slip a piece of lightweight fusible underneath, hand sew the seam closed, and use an iron to lightly heat the area to activate the fusible. I would use an ultra lightweight fusible. Pellon has some. MistyFuse would probably work too.
If your design allows it, hand quilting on both sides of the fraying seams will help stabilize them too.
For particularly bad spots, perhaps you can add an applique on top of the area to strengthen it. Alternatively, for a bad spot, remove and replace the fabric with a stronger one (sewing the seams by hand).
Fray Check dries hard. There is another product called Fray Block that is more flexible when dry. You could try putting a line of Fray Block along the seam on the fabric that is fraying; however, I doubt that would be enough by itself to stand up to the wear and tear a quilt takes.
Last edited by Prism99; 12-29-2013 at 12:12 PM.