Answer: Unfortunately, based on your description it sounds like your basil has been attacked by a nasty soilborne pathogenic fungus, basil fusarium wilt (Fusarium oxysporum f. sp. basilicum). Infected plants grow normally just as you describe until they are six to twelve inches tall, then they wilt. Symptoms can include brown streaks on the stems, discolored stem tissue, and sudden leaf drop. Mints can also carry the fungus although they will not show signs of it. This is initially introduced primarily through infected seed and then lives in the soil for up to 12 years.
The problem with fusarium in basil has been increasing in recent years; breeders have now developed at least one new variety, 'Nufar,' that has shown resistance to the disease. Other ways to deal with the problem include avoiding growing basil (or mint) in that spot again, and growing other varieties of basil besides the common sweet basil. Apparently some of the specialty basils (such as lemon and purple basil) have shown some resistance to the disease
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