Hi Alex, did you ever sort out what the issue is??
I seem to be haveing a similar challenge with the basil in my grow bed. The odd thing is there's a potted basil in soil three feet away that looks fine.
Alex - Do you have any webbing anywhere on the plant leaves? If so it could be spider mites. However, I'm currently having problems with thrips and the damage to your leaves looks similar to what I'm seeing on some of my plants. Look closely, do you see any long, slender, brown bugs crawling anywhere on the plant. Remember to also look on the underside of the leaves. Let me know.
So I had to kill off my basil plants :( When I pulled them out, there seemed to be something wrong at the base of the plant, like it dried out or something and worked its way toward the leaves. All but one basil plant was affected.
I had the same problem with my basil and lettuce. I think it's an airborne fungus. My problem was the humidity was too high in my greenhouse. I added a fan for air circulation, which someone here recommended, and haven't had the problem since with my latest crop.