Since your nails are driven by the power of your cuticles, once they're infected or damaged, so are your nails. When taking care of your cuticles, you shouldn't bite or cut them, as that breaks down their material and leaves them vulnerable to bacteria. However, cuticle clippers can be used for hangnails.
If they seem to be too high up on your nails, gently push them back, and be careful with hard tools like the one pictured. Finally, keep them moisturized like you would the rest of your body.