It sounds like a powdery mildew outbreak. It trim away as much as the affected parts of the plant as possible and then spray with a fungicide. Like this one: https://www.bonide.com/products/disease-control/view/777/revitalize-bio-fungicide-conc
Follow the directions on the label. That should stop the spread the disease. I wouldn’t consume the infected leaves. New growth should be OK. To avoid this on basil and other plants, never spray water…wet just the soil. Wet leaves are an invitation to fungal diseases. The high humidity and rain are bad enough…no point in adding to the problem