I'm not sure of the pest, but I am willing to bet that they came from the dirt. You should never ever use dirt in a hydroponics setup. You will get pests, disease, ph problems, and equipment clogging/failure. Only use a "solid" neutral substrate, like rockwool, hydroton, perlite, sand or rocks (only for certain types of system) which won't dissolve or muddy up the system.
My advice? Scrap all of the plants and dirt. Wash your system with diluted bleach and warm water (add a few drops of soap too) and rinse it well. Start in a clean area with new seeds or cuttings. You'll have a much easier time.
Good luck!

Here is a link that might be useful: DotCaDot - DWC Sweet Basil