Basil is very chill-sensitive, and the cold snaps we have overnight several times in the past several weeks may have stressed the plants or predisposed them to other problems. The wilting stem may be due to an infection in either the stem itself or the roots, as basil is also sensitive to being over-watered (roots kept too wet). Once wilted, when the soil isn't dry, it probably won't recover, but you can trim off the wilted stem if others in the clump are doing okay.

The brown areas on the leaves of the other plant might just be cold damage, which would explain why the younger growth on top of it is doing well. It doesn't look like an infection, but you can keep an eye out for Basil Downy Mildew, a fairly common and destructive fungal disease that some varieties of basil can contract (a few are more resistant). If the plant was kept indoors (if it was started from seed, for example) or was in a greenhouse before being planted, sun scorch might cause leaf damage like the symptoms pictured, though usually it's paler in color, having a bleached-like look. As the leaves adjust to the higher light intensity of being outside in the sun, the new growth will be fine.

In either case, the most heavily-damaged leaves can be snipped off, but nothing else needs to be done for now. To avoid over-watering, make sure the soil a couple inches below the surface is becoming somewhat dry to the touch before watering, which will help the roots get enough oxygen to stay healthy. (When they are too wet for long periods, the air in the soil pores is replaced by water, so the roots more-or-less can "drown.")

Miri