any way that you could set up a barrier on the ground so that the washed down seed can't wash into your lawn?

Or dig a diversion trench that allows the runoff to flow out around you.

First, seeds generally don't "wash" downhill as in runoff. They blow in the wind or get tracked there by people or animals.

Second, how do you know you are getting chemical runoff? The vast majority of herbicides are not mobile in soil. You are much more likely to get runoff from fertilizer than herbicides.

If your uphill "crabgrass" neighbor is truly taking an ecological stance, then an ecological argument might persuade him to address the problem. Their choice is ultimately resulting in more chemical use, not less. To minimize that, their best option would be to control the number of seeds leaving their property. The easiest way to do that is to mow more often. Seeds need time to ripen on the plant. If you mow them off regularly, you dramatically reduce the number of viable weed seeds.

billl makes some good points (as usual). Another point is if your lawn is dense and in good shape, you could dust the lawn every week with crabgrass seed and never see a problem.

Where do you live? (town, not zone)
What kind of grass do you have?
How high do you mow? and how often?
How often do you water and how much?