There is an easy way to grow extra Basil or propagate ( I think the term is ) , Do as I do and I have no problems at all, But I do live in a hot sunny place most of the year, doesnt get very cold, even in winter here, I live in sunny Nth QLD in Australia, and as most Aussies will know, it gets very hot here, not much cold at all, so the weather is perfect. OK, what I do is cut off the tips of your Basil plant, about 3 to 4 inches long, pull off the larger leaves, so the nuitrients from the soil dont go into the larger leaves, it goes into growing new growth, If you have several Basil plants you can do this with all of them providing they are established and big, then you can get heaps of these little beauties. Then I put them into a glass or glass jar of water, about 3 - 4 in each glass or jar, Then Wait!!!! it will take about a week or 2 for them to start growing roots, when their roots are large enough, then you can transplant them into pots with soil, keep them in a warm sunny spot, and they should grow like wildfire. I have never had a problem doing it this way, and in our winter months, I just bring the younger ones inside for a while, I put them outside during the day, so they can get some sun.
Keep them well watered, when they need it, and all should be good. Another thing I do is pluck off the seeds or flowers if they are to young, as I find if you want a good bushy plant with lots of leaves, then pluck off the flowers, so they dont go to seed to early, wait until they are a mature plant before you allow this to happen.
Hopefully this has helped a bit and given you some new ideas.