I get white fly ... like crazy...on basil. A few safe things to do:
Hang the yellow sticky tags to trap them. They work if you keep agitating the plant to get the bugs moving.
Spray the plant gently under the tap. The flies (If that's what you have) will go elsewhere for the moment, but then you must immediately move the plant to a spot where the flies cannot find it. They don't care for the less juicy plants. I do this with my huge hibiscus each Fall and it's finally, the only thing that works. I bring it inside instantly after spraying it right at the back door. Good luck to those little buggers trying to find something to support them outside all winter. (maybe that's why the squirrels were a little itchier this winter)
Basil can be difficult; I guess it's more than just we who love it.