To prevent infection from a cut on your finger, you should practice basic hygiene by washing it with mild soap and keeping the wound covered with a clean bandage. Only change the bandage once a day unless it becomes dirty. If the redness or swelling increases, it is advisable to see a doctor promptly. If you cut yourself with a sharp object and your last tetanus vaccine was more than 10 years ago, it is highly recommended to get a booster shot.