Make sure the wound is completely cleaned out, you don't want that getting infected. Wrap the cut in gauze dressing.
If you are positive it is a cut vein, don't worry too much, veins can heal themselves. But make sure it isn't are artery, that can be serious. Check to be sure you didn't cut a tendon. The way to find out if you cut a tendon is if you can't point your finger straight. If this happens, you need to see a doctor, that will not heal.
If it is still bleeding, which means its been bleeding for 5 hours now and you should be at the E.R, apply pressure to the base of the finger until it stops. Finally, if the wound is so deep that it doesn't heal on its own, you may need to get stitches to close it. Just keep an eye on these things to make sure it heals properly, if now of the above happens it should be fine without getting doctors involved.
