Hi,
Don't worry it is easy to get basil seeds : )
You do not have to let the plant die, just the flowers. The flowers will eventually die and make tiny capsules and the top of the stalk where the flowers are and the capsules will start to dry. Once they become brown and papery, there will be dark seeds in within them. Having a small bag handy to cut the dried flowers into works well. Then you can sort out the seed from the dead flower on a plate or piece of paper.
Oh something important! You do have the plant outside? If not, place it outside. Basil needs to be pollinated by bees or you won't get seeds to develop.
Remy