I grow basil inside my window. Now that the growing season is ending and there are flowers at the tips, does anyone know how to save/preserve these buds in order to start a new plant next spring?

Basil is an annual plant, so you need to let it go to seed, which means the seed pods will turn brown, open and scatter the seeds.
Best way is after the pods turn brown, make a little sack w/ gauze or cheese cloth and thread.
Or you could watch until the pods start to open, cut them off, open them up and get the seeds.
Sort of a pain, basil seeds aren’t too expensive. But the pods must be brown.

What I do is wait until the pods dry and then run my fingers up either side of the seed pod to pop open and scatter the seeds in the pot. Then the new crop sprouts amid the stalks of the old. I am on my fourth generation now I think.

These are both such great answers – thanks, you two! But I must really not be understanding your terminology so please explain: The flowers at the end of two of the tips – are those what you are calling “pods?” Right now they’re not in the earth anymore – the two stalks the flowers are at the end of are in water. Should I take them out of the water so that the flowers will turn brown? “Pods” to me signifies something flat and large. These flowers are diminutive and dainty so perhaps I don’t understand what pods refers to. Last, I don’t know what climates you are in but I’m NY where the weather just dropped from 75 to 55 degrees. When you say “scatter the seeds in the pot” do you mean next spring? Or now? (I have the basil inside a south window. I was hoping it would last the winter.)

Sorry, I live in Florida so I can keep basil outside year round.

Once the plant has finished flowering little seed pods form and there is a tiny black seed inside. Here is a really good picture of it.