Definitely not sunburn which makes pale areas not dark spots.
We are thinking maybe basil downy mildew. If so the underside of the leaves would look about like this:
Dark dots but mainly on one side and mainly between the veins.
If that is the case, then unfortunately yes, you have to dump them. It is a deadly disease with no cure:
https://www.thespruce.com/what-to-do-fo ... ew-1762532
Remove Infected Plants
Basil that has become infected with Downy Mildew may have a yellowish appearance, similar to a nutritional problem.
The leaves turn yellow and fall off, while the stems wither, and the plant eventually dies. The most common symptom is the dusty spots, actually spores, that are located on the bottoms of the leaves. If you want a definitive diagnosis, you can bring an affected leaf to your local County Extension office for identification. Carefully pinch off a leaf and place it in a ziploc baggie for transport.
Once you are certain that it is Downy Mildew, cover the entire plant with a bag to prevent as much of the spores from being shaken off as possible. Pull up the entire plant and remove from garden area. Do not try to compost or try to remove only the visibly affected leaves. Both of these techniques can result in the spread of the spores. Do not save your basil seeds from any part of that year's crop. The spores can infect the seeds and affect the following year's plants.
Will It Stay In My Soil?
The disease can stay viable for many years in the soil. Some experts say that the spores can stay viable for 8 years or more. If nothing else, rotate your crop of basil and plant in an entirely new location for the next season.
Very sorry to be the bearer of such bad tidings.
Basil downy mildew has become an endemic problem in some areas, so much that people (like imafan) can't grow basil at all.
Here's an article about the history and spread of this disease: https://vegetablemdonline.ppath.cornell. ... Downy.html
So far this year basil downy mildew has been reported in
Florida, South Carolina, Ohio, Louisiana, Pennsylvania, Massachussets, Connecticut, Georgia, but it is likely in other places as well and just hasn't been reported.