Diseased Basil

Can you possibly look at these pictures and help me figure out what is wrong with my basil? It is being grown inside a greenhouse in gallon pots. The healthy leaves start to turn brown and eventually shrivel up. Is it likely fusarium? It even has been happening on my Nufar variety plants.

It is hard to tell from the pictures what you have, but it is not likely to be fusarium. When fusarium strikes, almost overnight the leaves wilt and drop off as the disease cuts off the water supply to the leaves. Typically, the leaves are green, but completely flaccid, and then after a few days they wither away.
In your case the leaves still have some turgor despite the progress of the disease. Whatever is attacking the leaves is doing so section by section.
Another common disease is gray mould caused by Botrytris cinerea. But the brown spots appear to rule out gray mould in your case.
I suggest that you take a sample of your leaves to your local extension office for diagnosis. Many local extension offices offer diagnostic services to growers. Nutritional problems and mite infestions can produce symptoms similar to yours so these possibilities should be checked too.