It's not clear in the post -- is her whole fingertip red "from tip to first joint" or is it just red at the cuticle itself? The former, the way I read it.
If ONLY the cuticle is red, slather in Neosporin all weekend and keep it covered with a band-aid and change both after every handwashing, and then see a doctor Monday. If it swells at all or gets very tender, get to a doctor over the weekend.
But if the ENTIRE fingertip is red right now, and hurting? Get her to an urgent care center first thing tomorrow; she probably needs antibiotics. Don't fool around with this. She could have an infection that is spreading.
A few years ago I had the tiniest spot on one finger, like a pimple, with a white head and the smallest pinkness around it. No biggie, no pain unless I bumped it directly somehow. My elbow and upper arm on the same arm were achy. I went to the doctor about the elbow only and didn't think to mention the finger. She spotted that "pimple" and immediately put me on antibiotics. The spot on my finger didn't look all that bad but actually was infected enough that it was the cause of my whole arm's ache--an infection was spreading. Don't ignore even a tiny, fingertip-sized problem!
If you go to a doctor and they say it's nothing, by the way -- better to do that than to wait and find out you should have gone sooner. Sometimes moms on here get a case of "I don't want to waste a doctor's time" or "I don't want to look silly." Better to look silly a time or two than to not go the one time something really is a problem.