Spray basil plants with a strong pressure water sprayer each evening for a week to remove aphids and whiteflies. A spray of water often is enough to get rid of the pests.
Mix 1 part insecticidal soap with 10 parts water in a sprayer and add 10 drops of neem oil. Shake the sprayer well to combine the ingredients.
Spray the basil plants, paying particular attention to the underside of the leaves where pests like to hide. Spray at dusk to avoid the soap burning the leaves in the sun.
Repeat daily for a week. Spray weekly thereafter as a control measure.