Basil cuttings root very easily in water. Just change the water and rinse off the roots, every few days, to keep the environment clean. It’s not fatal, but algae competes with the basil for nutrients, so try to avoid breeding algae.
While plants are still healthy (September in northern gardens), select stems that are free of any pest and disease. Remove any blooms. Make a clean cut just below where the stem branches and take off most of the large leaves. More leaves (and blooms) add stress to the cut stem trying to survive, but leaves need to be kept to photosynthesize and feed the cutting. Crowded leaves can also lead to mold. Roots will appear in 3-5 days.
In order to enable photosynthesis, give the cutting lots of light. The photos are of cuttings on a kitchen windowsill. The plant on the right was left to see how long it would keep growing in water. It kept growing.
In order to raise a good healthy plant, pot the cutting into soil when roots are about an inch, which might take 2-3 weeks. It doesn’t really pay to leave them longer, because longer roots are easier to damage during transplant. Just like tomatoes, the deeper you plant the stem, the more roots the plant will produce along the buried stem. How deep you want to plant is up to you, it will grow as long as the existing roots are buried.
Basil does not need a lot of fertilizer, but when you do fertilize, use something rich in nitrogen in order to maximize leaf growth.
One cutting in a 6″ pot works well. The potted plants sold in grocery stores contain as many as 20 seedlings competing for nutrients and the crowding leads to disease. They’re not meant to last, just to give you a nice bunch of fresh basil.