I think I've figured out how to successfully root basil cuttings.
Basil plants normally like loads of direct sun and plenty of water. But cuttings don't seem to like this. My cuttings seem to wilt very quickly and often become pitted with large holes. They grow decent roots and they recover reasonably well once planted. But it takes time.
This is how they normally look. Here's my most recent planting which, even after a few days in soil, looks pretty much the worse for wear.
But lately I've been giving my cuttings indirect sun, mostly semi-shade. It seems to be working. This one hasn't yet sprouted roots, but he's been in the water for nearly a week and he's still looking as healthy as he was when I cut him.