Basil requires well-drained moist soil. If the soil does not drain well, the water pools around the roots and the basil plant will get too much water, resulting in wilted leaves. Likewise, when the soil dries out and the leaves do not get enough water, they will wilt and shrivel.
Slugs and aphids are basil predators and will cause plant leaves to shrivel and wilt. You can tell if slugs have gotten to the basil plants by the presence of leftover slime trails on the leaves. Keep slugs away by mulching with bark. Aphids will infest a basil plant; eliminate them by spraying the bugs with insecticidal soap.
Basil requires hot summers and consistently warm temperatures. The first sign of frost will completely kill a basil plant. In areas with cold summer or fall nights, bring the basil plants indoors to protect them from the cold weather.