Well, I think I know what you mean. You get these cuts around the base of the nails, I guess? Because I get that too, a lot. Never infected, though, but frequently... Inflamed, I guess you can say? Swollen and red, that is.
Anyway, if you want help to deal with them, the only advice I can give you is to cover them up. I often unawaredly start rubbing these sores, which only makes them worse of course, and fingertips gets a lot of wear during the day anyway that you never notice. They can stay for a long tine if I leave them open, but just a simple bandaid over the fingertip usually makes them heal up over the day or the next.
As for how you get them, it's worst for me during the winter (when it can happen several fingers a week if I don't take care) when I don't wear gloves, so it's obviously related to the cold someway. Most likely, I guess, it has to do with the nail-bands getting dry, then stretched, and then they break. Kind of like lips tend to do in cold weather for some people as well. So maybe you could try using some sort of moistening cream on your fingers or something? Also check what kind of soap you're using, that might be part of what's drying them out.