The Greek Columnar Basil is growing in a container, sitting outside at low 40's at night. There have been lots of aphids activity which I treated with blasting water, and diluted neem oil. Does it look like Downy Mildew or Sooty Mold? Should I toss it in the trash, or remove sick leaves and try to save it?

If you can't be certain it's not a disease/bug, then probably best to toss it. If it's near & dear to your heart, snip off a clean-looking top and bring it in for rooting, but if it gets infected, toss it.

I never have problems with diseases on my basil (though I know some places have problems, since many varieties are sold as "resistant"), but something that happens with all of my basils when it gets cold, but before frost, is black spots. The weakness may make them prone to aphids, though I have never had aphids on basils, indoors or out. Not sure what it is, but since you said that it got down into the low 40s, that might be the problem, since that's about when mine start getting the spots. They are the first plants I lose every fall. I usually take cuttings in late September, before the spots, to clone for the hydroponics.

Thank you both. I have some clean cuttings growing indoor since Oct. I think I'll have to toss this before it spreads to my African Blue Basil. I was testing their cold tolerance as perennials.

I have seen the end-of-the-season spots on my monarda and mint and maybe some other stuff. I don't know if it's disease or the plant just going dormant.