For cracks in a concrete driveway, see "Concrete cracks." For small cracks in asphalt, buy a tube of asphalt crack repair compound, sold in a tube tha's meant to fit into a caulking gun.
Sweep loose dirt and debris out of the crack. Then apply the asphalt repair compound according to the label directions. For large cracks in asphalt:
1) Use a broom or stiff brush to sweep out cracks. If you have a shop vacuum, use it to vacuum up loose dirt and debris.
2) Use a cold-process asphalt repair compound to patch holes and large cracks. Follow the directions on the label.
Note: You can also ask your home improvement dealer whether they carry a self-adhesive polypropylene rubberized fabric, sold in rolled strips, that you just cut, peel, and stick over a crack or small hole.