For some reason, your flowers did not get proper pollination to set seed. So the flower part dies away instead of forming seed.
Remy

That's strange, since I see bees visiting the basil flowers. Is there anything else I can do to help the polination?

My thought would be some kind of insect on the plant. Though I have never had insects on my basil only slugs tend to attack the bottom portion of mine. If not an insect then maybe lack of water, a plant needs the same care during seed development as during bloom and regular growth. I only say those options cause you said you have plenty of bees visiting, and I have never had a problem with my seed not forming on my basil and I never pollinate mine myself.

Remy is a very smart person and believe me she knows her pollination techniques. With that said if you think that pollination is the problem then you can encourage it by taking a small paint brush and rubbing it from the outside of the bloom to the inside (stamen to pistil).

It is not only whether the bees visit. They of course help. Weather plays an important part. If too hot or cold, or too wet or dry, failure can occur.
Your flowers now may set seed though the earlier ones did not. This happens a lot. Let say the last flush happened during a heat wave and all the flower heads died, but the temps dropped a few degrees during this flowering set, you will see seed this time around.
Remy

Thanks for all the replies. I decided to uproot the plant in any case since I needed the space, but did first pull off individual branches to look at. Lo and behold, I did find some mature seed pods! I managed to gather 20 odd seeds, which is more than enough for my modest needs. I guess some unfavourable condition must have caused the low success rate. Could it have been because the first flowers set in the middle of winter? Winter here is fairly mild (never below freezing), but it can be overcast and rainy for weeks at a time.

Something that did interest me is that several of the mature seed pods had less than four seeds. Is this normal, and should the seeds that I did harvest from those pods still be viable?

Thanks
Neilen

Glad to hear you did get seeds! It doesn't matter how many seeds there are in pods(this goes for all plants) as long as they are plump little seeds. Your weather probably was a factor in the seed development. Years with long periods of cool rainy weather causes lower seed counts with many of my spring bloomers, and I assume your winter weather would do the same.
Remy