I used to do this, too. It would be so bad the tips of my fingers would be bloody and raw sometimes. For me it is a nervous habit, and I still do it occasionally - just not to that extent.
There are two things that helped me immensely. The first is, as others have recommended, keeping your hands & fingers moisturized. Bits of dry flaky skin would start me picking, which turns to biting and just goes downhill from there. I love Burt's Bees lemon cuticle butter. I also keep a bottle of lotion handy near the sink and on my bathroom counter and desk at work, and use it often.
Second, the thing that helped me the most is wearing false nails. I wear acrylics off and on for about the last 15 years. I keep mine very short and natural looking - I have a very good nail tech and most people think they are my real nails. I never chew or pick my fingers when I'm wearing them. I am not sure why, I just stop cold turkey when I have them on. I really recommend trying gels or acrylics if you are not opposed to it. Nowadays you can even get the OPI Axxium gel manicures, which is a colored gel overlay (almost like a long lasting manicure) over your natural nail.