I use Ranger distress inks, and my Stampin' Up! inks and a Mini-Mister spray bottle from Ranger. I put a dropperful of distress ink in the mister and fill the rest of the way with water. I can usually dye about 3 yards of binding with one mister bottle. If you have a non-stick craft sheet, just lay the seam binding on it, and keep spraying and moving it around, sopping up ink from the craft sheet, etc. until the desired color is achieved (but in most cases it dries lighter!) Then I push the binding into a plastic kitchen bag, get all the air out and then I just tie a knot in the bag. I leave it like that for 12-24 hours, then take it out and lay it out somewhere to dry completely. Every now and then I just give it a fluff as it dries. I've also wrapped the binding up in tin foil after dying it, and put it in a 400 degree oven for 20 minutes to dry. I do several at a time, as I don't like heating up the oven for just one bundle. The only thing with this is that it dries so quickly the dye can be splotchy and if you wrap it really tight, it'll be really crinkly. I have a big basket full of the binding I've done. I love to just dig through it... it's all so pretty!
Hope that helps!