I don't really have any good ways to fix it. You can put some Fray Block on the run & maybe see about adding some stitching or applique over it to hide the run if it's very noticeable.
You've gotten some really good advice already about preventing it. The other things I'd suggest is considering your needle & thread. I'm not sure if you are doing the Quilter's Knot after hand quilting or applique as I use that knot for both. I use a 50/2 cotton for applique & for that I do wrap around my milliner's or applique needle 3x -- 4x for looser weave fabrics, or 2x for batiks -- before pulling through the background. The motion in that case is pulling straight down.
For hand quilting, I'd go with Jan's method. With 50wt thread, I'd stick with 3 wraps unless you're either working with a batik or a very fine needle (like the ThimbleLady needles). With 40wt or 28wt thread, I switch to 2 wraps. If you're not already working with special Hand Quilting thread, a thread conditioner like bee's wax or Thread Heaven will help. If I have an especially stubborn knot, sometimes I'll rub a bit more bee's wax on it to help it slip through. Or I'll just move my needle over one thread's width & try again. The last thing I can think of that could be causing the runs is if you have the fabric a bit too tight in your hoop. You want it to have some give during quilting -- and especially when burying the knots.