Remember when I started those basil cuttings? It was 5 weeks ago. You may see it here.
After 4 weeks first tiny root appeared and frankly I was not sure whether it's a root or not. Today, 5 weeks after I am sure my basil cullings are rooting!
In the meantime I was noly adding water, not exchanging with fresh one.
My recommendation is clear. Propagating basil from cuttings works. If you can do it, still this autumn, save your overmatured basil plants by taking cuttings. This way it's possible to enjoy your own basil entire winter.