The disease infecting your basil plants may be leaf spot caused by fungus, Colletotrichum spp. High humidity and wet leaves favor disease development. Also the disease mainly spreads from splashing water. The main symptoms are appearance of brown to black spots on leaves. As the disease progress, this spots coalesce and dry out and eventually results in leaf fall.

Management:
1. Remove the infected leaves and burn them.
2. Avoid overhead application of water and wetting of leaves. Apply water directly to soil.
3. Also isolate the infected plants from others.
4. If disease is severe you can spray fungicide which contain potassium bicarbonate.