Probably sunscald. Lately it has been WAAAAAY too hot to grow just about anything outdoors. Basil likes temperatures of about 75F-90F and it has been a LOT hotter than that for at least a month.
Those little black specks are either feces or fungal spores. When a plant leaf is sunscalded it is in a much weakend condition and any insect that flies by is attracted to it. Just pull off the affected leaves. Same thing with fungus, a weakened plant is more susceptible. To alleviate sunscald damage use a shade cloth or move the plants to a place that doesn't get this blistering heat.
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