Your basil plants may be suffering from a particular disease that is virtually impossible to control, known as Fusarium wilt. Once it is in the soil the best thing you can do is try planting basil in another location, or to search out a variety called 'Nufar' which is bred to be resistant to this particular disease. There are other diseases that can affect basal but the best thing you can do is to try to provide proper care meaning moderately moist soil, moderate fertilization, and plenty of sunlight. The less often you water the plants, that is the less you keep the foliage wet, the fewer disease problems you'll have to deal with on basal.
The following website provides more information on diseases and pests of basil, and how to grow it successfully.
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