This is a new insect for me, I had never seen anything like it before. This is Cardinal Basil, grown from seed and it was a perfectly healthy plant until last week. Blotch Leafminers are very different from the traditional serpentine leafminers I have been plagued with before.
http://blogs.ifas.ufl.edu/monr...

Yet another Corn pest... sigh. We are bound to see stuff like this become more and more widespread as the Devil's weed dries up our aquifers and stinks up our fuel tanks.

From literature it seems to occur in periodic mass outbreaks in Florida, so this is most likely what you are facing right now.
To the best of my knowledge the only effective remedy for corn is clothianidin, but this is used only during mass outbreaks and makes zero sense for home-scale basil gardening.

Unfortunately basil is a pest magnet, and the only 100% sure cure I found for it so far is to grow it under a very fine mesh covering that will prevent bugs from laying their eggs on it. Just make sure that mesh is well anchored to the ground, otherwise cutworms will find their way inside.