Answer: They may have been insect eggs, but unfortunately it's pretty tough to make an identification long distance. Here are some photos of insect eggs to show the type of form they can take.
There is no "common" basil pest that would do this in particular so I don't have an immediate suggestions for it. As long as you were able to remove them you probably don't have to do anything more. If you see it again you might ask your county extension if they can identify it for you.
In the meantime if you see any insect damage on your basil you will want to make sure to do a thorough cleanup of the plant debris after frost this fall just in case it is something that might overwinter and then recur next year.
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