It depends on what's causing your dry skin. For example, if you wash dishes frequently in hot water, wear gloves. Or if you often wash your hands with harsh, drying soaps, switch to a mild soap if possible and rinse well.
Begin healing your thumb tips by sealing the cracks with a liquid bandage. Use a moisturizing lotion on your hands several times a day. It's best if you use lotion while your hands are damp from handwashing. Use a thick lotion, such as CeraVe, Eucerin or Cetaphil.
It also may help to treat the affected skin with a heavier, oil-based cream or petroleum jelly (Vaseline, Aquaphor Healing Ointment, others) while you're sleeping. As part of your bedtime routine, apply the product to the thumb tips. Then cover them with gauze secured with tape or wear cotton gloves.