Hydroponically Grown Basil - How can I keep it going?

It doesn't always work, but often enough, as Nozlee said, field-grown basil will take root in a glass of water and last far longer than it otherwise would. Cut off some from the bottom, stick it in water, and wait. Your hydroponic basil should already have roots, so you're ahead of the game.

If you soak a basil stem in water, it will start to produce roots. Then you can plant that root into soil and the basil will continue to grow!