Although I prefer to start basil seedlings in our AeroGarden Deluxe using the accessory kit, once it is big enough to start again from cuttings, I take the parent plant out to finish its life in a glass of Hydroton grow rocks:
Here are a few of the cuttings I rooted from stems cut off the parent plant shown above:
The leftmost container is a beverage glass filled partially with sand, with a shallow layer of Hydroton grow rocks on top.
In the middle is a mason jar. The insert is made from a Kozy Shack individual pudding cup. Because the glass jar is clear, you can see the basil roots growing down through the holes I poked in the bottom.
On the right is a container made from two juice bottles I cut down and nested together. The liquid nutrient solution flows over the clay pellets which line the small inner container and comes out through holes I punched in the bottom. Then it floats back up to fill the outer container.