I'm looking for some advice on what I can do to prevent runoff water from going into my neighbor's yard. I am an auto detailer so I am constantly washing cars in my driveway. The neighbor also parks his truck and RV camper in the dirt section of the yard. Here is a picture so you can see what I am talking about. :-)
..Your driveway would be the cracked concrete slab to the left, I assume?
If the picture is not misleading, there appears to be some slope towards the street. If so, rent a concrete saw or buy a concrete blade for a saw you own (but unless you own a water-cooled saw, a rental saw that is water-cooled will generally do a better job, faster) and cut a bunch of grooves 1/2" (or so) deep leading to the curb/gutter at streetside.
That may be all you need to do, or you may need to make some sort of raised edging at the right side of the driveway. That would probably best suit the area if formed from concrete, but something like a rubber glue-down garage door bottom seal (the type that's stuck to the floor, not the type that's stuck to the door) might be quicker and cheaper, until it dies in the sun and needs to be replaced.
You could offer to install a trenchdrain into your neighbours sand. depending on your runoff, that should solve the problem.
unless you want to go with the above solution and build a mound, which may or may not be a better solution depending on the way the water runs, i.e. you dont really want the water to pool on your side.
Epoxy a bunch of bricks and/or pressure treated 2x4s to the driveway. The climate is super-dry, and because of the pitch of your driveway and the amount of water on it, you only really need to contain about a 1/2' high wall of water, and it doesn't matter if it leaks a bit as long as most of the water is directed down the drain.
For the truly cheap, laying out a bunch of pressure treated 2x4s along the edge is likely to work; put them back in the house when you are done.