Corinna at my local extension service found the answer to my problem. Thanks, Corinna!. The attached link shows a picture, and suggested solutions. Although my basil plant didn't look exactly like the picture a couple days ago, today when I checked on it, the brown circles had become more defined, and clearly looks like what is in this link.

Here is a link that might be useful: Hort.uconn.edu

I can see the four lined plant damage in that photograph, now that Corinna was able to ID it for you. The image wasn't really helpful to me before. There have been a lot of postings about this particular pest this year. They LOVE herbs.

Be sure to stay away from the strong pesticide solutions cited in the fact sheet.