Cut flower stems several inches below the bottom seed pod.
Gather the flower stalks by the stem and rubber band them together.
Place them, seed heads down, in a paper bag and secure the bag around the stems. Hang the bag from the stems and allow them to sit for a couple of weeks in a dry place.
When you're ready to process the seeds, shake the bag a bit to knock any stubborn tiny seeds from the stalk. (If the basil seeds dried on the plant, you can skip that step and go straight to the next one!)
Line a rimmed baking sheet or casserole dish with a sheet of white paper. This will help you see the tiny black seeds as you sort the seeds from the chaff.
Carefully open the bag and lift the stems out and give them a last-chance shake over the white paper. (Compost the stalks.)
Pour what remains in the paper bag onto the white paper. Blow very gently across the pile and watch as the dry chaff lifts away, leaving behind the basil seeds you can plant next year.
Store fully dried seeds in an airtight container in a cool spot.
Notes
I recommend you do this outside, so you don't end up with dried basil leaves all over the floor!