Injuring the tip of finger may or may not affect one's fingerprints. Based on the size of the injury
you describe, it is likely that the fingerprint will grow back although it may take some time. The overall size of the injury is not necessarily what will determine this--the depth of the injury is more likely to affect your future fingerprint.
However, as with any injury the most important thing is to make sure that you are evaluated by a physician. Depending on the mechanism of injury, you could be at risk for wound infection or an infection related to the injury itself, such as tetanus from a puncture wound. Even if you don't require stitches or antibiotics, it may be a good idea to follow up with your primary care physician
to make sure there are no concerns about wound healing or infection. He or she can prescribe antibiotic ointment if needed or give recommendations on how to protect the area with wound dressings or bandages.