Those are blotch leaf miners, a tiny larvae that feeds on the interior of the leaves. New foliage might be able to be protected with occasional applications of neem oil. Snip off the infested leaves and discard.

I've the exact problem. Can Confirm, as secretmail021 has identified, it is a very very very small insect. White-green color, approximately as wide as 1-2 human hairs, about 1.5 mm in length... you won't see them on the leaf, Instead, pick a leaf and shake/tap it over a well-lit preferably dark surface.... if they fall off, you can see them crawl. Happened with indoor potted plant that was otherwise healthy for the last month. No idea for the solution, but act fast, because it spread from a few just a few leaves to a few plants over the course of half a week. RIP basil.

I just had to get rid of 7 basil plants and their soil due to Downey Mildew. The leaves starting turning a blotchy yellow with brown undersides with tiny spores. It's a relatively new development in the East since 2008. Check the underside of the leaves if they are normal, it could just be that they're getting too much rainy, cool weather.

Very possible that the little black fellows are aphids. If so, they suck plant sap and will weaken the plant. Rub them off, or use direct hits of insecticidal soap, diluted according to label directions. In both cases, repeat as needed. I'm not able to see enough detail to suggest what the white might be.