Have you given it any extra food? It sounds as though it might need a dose.

If the leaves begin to droop and fall off, especially if it's getting cold where you are, pull the whole thing up and forget basil until next spring. It could simply be dying of old age brought on early because of the cold. Basil can take an awful lot of rain, as long as there is fast drainage, but it can't take the cold, and isn't too happy in the wind either. Depletion of the soil doesn't usually happen so quickly in a garden, however - so I'm guessing your basil is in a pot. If it's in a pot, definitely feed it.

Thanks for the advice. It is in a pot, and I did add a small amount of houseplant fertilizer on Sunday. No perceptible change yet. Maybe I should add some more?

I takes awhile to get a response from fertilizer. Did you use something liquid.... like MiracleGro or did you sprinkle on something dry? Liquid, properly mixed, will get the fastest reaction.

I used some very dilute liquid miracle grow. I will watch it for another day or two (?) and add a bit more if I haven't seen a change. Thanks again :)

Don't overdo the fertiliser! You can kill it with overfeeding, especially when the plant is very sick and won't want to eat much. Follow the manufacturer's instructions as to amount to use, and how frequently. It can take a couple of weeks for the plant to pick up after a half-drowning or other trauma!

The lack of iron also have has an effect on leaves. Suggest that you mix up a batch of ironite and apply that to your soil. Be sure to use the proper amount and wait at least 20 days before seeing if it has helped to reduce the yellowing. Iron sulfate takes a while to get to plants, but is carried there by natural nitrogen. Here, I apply this every spring and its been very effective in giving me some very green, healthy plants.

Thanks again for all of your suggestions.
It has perked up considerably and is getting much more green. I think I may just make my very first ever batch of pesto this weekend!