Place the basil leaves in a clean sink or large-enough bowl to contain all the leaves. Slowly run cool water over the leaves until they are mostly covered. Let them remain in the water for several minutes.
Place your fingers in the bowl and gently swish or swirl the leaves around. If you notice any obvious dirt, focus on that leaf or stem section. Check each of the sections and individual large leaves, especially underneath, where dirt sometimes splatters upward from garden soils. Rinse them around, empty the water and add more. Repeat until all dust, dirt, debris and insect matter is removed.
Pour the water from the bowl or drain the sink and lift the basil out onto a clean kitchen drying towel. Layer the basil loosely across the towel, covering it with another clean towel. It will drain somewhat rapidly, then gently dab the leaves to remove any excess or remaining water. Alternatively, you can leave the top towel off and just let it air dry. Avoid crushing leaves prematurely, before you actually want them crushed or chopped for recipes.