Suckers often form after there has been some damage to the roots - did you recently repot this rose?
In my experience, suckers from below the graft have never been worth the trouble to grow on. The rootstock exists to allow roses to be propagated true to type and often to add vigor to the rose grafted onto it. Some rootstocks are hardier than others while others are more disease prone. This site has an interesting, short discussion on rootstock roses if you're interested in learning more about them.
In my case, I've seen the rootstock Dr. Huey grow from its roots into a six foot tall, gangly and very prickly monster, with insignificant, ugly flowers. The grafted rose then died (possibly, I thought at the time, because Dr. Huey had appropriated all/most of the nutrients and water for itself), leaving me with a nasty mess. In another case, the rootstock suckered into a shorter impenetrable and prickly shrub that hid the grafted rose. I was able to save that rose by viciously pulling out the suckers.
My advice is to remove the sucker ASAP - but DON'T cut it. This WikiHow site contains decent instructions on how to excavate around the sucker to expose the root, and then how to pull the sucker from the root, at the root. I should note that I'm not usually a fan of the information on that site, but these instructions are easy to follow and correct.
I would not recommend that you root the sucker itself, as rootstock roses are not grown for their flowers.