If the problem drainage is contributing to washout in the form of a gully because the water collects next to the driveway in one place before flowing into the backyard, you can install a simple plastic catchbasin and drain line or drywell. Much of this depends on what kind of energy and equipment you have available. Most home improvement stores carry these plastic (HDPE) catchbasin assemblies and the appropriate pipe to go with them. Install the basin in the ground right next to the driveway, at the low point where the water collects. The dig a trench for the pipe to a place where it can day-light and drain away. The pipe needs to slope away from your basin. If you have no place for the pipe to go, then you might try testing to see if the sub-soil next to the driveway is gravelly and free-draining (if you pour water into it, it disappears into the ground quickly). If you're lucky, then you might be able to construct a drywell at the low point next to the driveway using the same plastic basin, without pipe or bottom in the basin. You'll have to dig the hole under the basin large and deep enough to place 3-4 feet of crushed stone into which the water will drain and subsequently perk into the ground.