In wet weather, slugs may come out and eat your basil plants. They feed during the night so you aren't likely to see them, but their slimy trails should be apparent.
Japanese beetles are mostly located east of the Mississippi River. They damage basil plants with holes of varying sizes and shapes in the leaves and chew marks along the edges.
A very common pest for the basil plant is the aphid, a small insect with long antennae. Aphids prefer to eat new growth, causing newer leaves to be the wrong shape and smaller than usual.
If your basil plants have spots on the tops of their leaves, you might have leafhoppers. Although they don't eat basil, they cause cosmetic damage and can carry diseases that further damage your plants.
If your basil is wilting and yellowing inexplicably, nematodes might be the culprit. These microscopic worms attack the root system and cut off nourishment to the basil plant.