That's a very localized effect. If it were a watering issue the whole plant would look worse. But I'm wondering about the leaves on the ground as well.
I agree they look burnt. My best guess is that they weren't harden off before going outside. Did you gradually introduce them to the outdoor sun over a week or so? More information about their history would help.
I'd ask if they got a little frost but you're in Houston :).
In any event the new growth looks good and from what I can see of the soil it looks damp enough. If the soil is moist they are getting enough water. Basil likes moist but not wet. I think whatever happened is in the past and they will be OK.