Black Magic Marker or Sharpie.

SoSoft Fabric Paint by ArtDeco

Are they indoor rugs? Do you plan to have them for a while? If yes, any large carpet place can redo binding and not terribly expensive. Perhaps you could request a polyester binding that will stand to sunlight better than black cotton. Again, not expensive but moving furniture off rugs, and transporting back & forth probably a huge pain

Fabric Paint or have it bound by a professional.

There are fabric paint markers ... saw them at Walmart.

Any black permanent marker would do. And some brands of fabric paint brag that they dry "soft".

Thanks for your help. Markers would never work because these are BIG rugs, and I would probably be asphyxiated! I will look for the SoSoft fabric paint. If you have any further ideas on how to tackle this, I'd appreciate it. The rugs are 11X14 and 2 5X8's. Oh am I going to have fun!

At the risk of sounding cavalier with someone else's money, have you priced replacements? IKEA sells sisal rugs with binding that are super cheap. You can also check some of the online sites--Overstock definitely sells these at a big discount. If the rugs are old enough that the binding has faded, maybe it's time to just replace them instead of fussing with a lot of work and/or expense to make the ones you have look like new again.

i had the same thing happen to a rug that gets strong sun coming through the windows... i asked at the rug store if it would fade and he told me it would, and it quickly did! (they're only 1 1/2 yrs old and one side is as you describe) too bad they don't bind with a more sun safe fabric!! i bought it anyway because i liked the rug and the price...
only one side is faded badly and i quickly went over it with a sharpie side to darken it up abit and make the fade less noticeable... i also won't have this rug in the current room much longer, so haven't turned it for more even fade. if you have all the rugs to do entirely, maybe certain fabric paints would be easier but i have never used a fabric paint that i would want to use on the rug tho... i would probably be more tempted to try a watered down acrylic paint myself i think!!
but, sharpies are wonderful fix-its... every home needs a nice selection of sharpies!!

And they do make BIG Sharpies.

The SoSoft fabric paint is an acrylic.