Where are you located? They look like scale insects to me but I've never seen them on basil here.
I do get them on my overwintering indoors citrus. Last year they got on parsley that had sprouted in the citrus pot so I guess they do attack softer/more tender plants, but usually I see scales on harder surfaced plants including orchids, but mainly citrus, bay laurel, hibiscus, etc. I've had Roselle that got covered in them and cotton, too.
For minor/early infestation you can wipe them if with cotton swabs dipped in alcohol. Let me see, in UK, they are called something buds and something spirits...hmmm can't remember right now.... Hopefully basil is not too delicate for this treatment.
ETA (Edited To Add) -- found the ref