Bring potted basil indoors and place the plant in a warm sunny area as soon as possible after frost exposure. Keep both indoor and outdoor plants sufficiently watered, as damaged plants and dry soil may result in too much stress for the basil to survive.
Trim back the dead and damaged leaves and stems to the nearest healthy leaf node. If the entire top of the basil is damaged but the stems near the base are still healthy, cut back the basil to within 2 or 3 inches of the base. The plant may grow back from the healthy stems.
Fashion a frost covering for outdoor plants if the damage occurs at the beginning of the growing season. Place a purchased plant cloche over the basil when cold temperatures are predicted to prevent further frost damage. Alternatively, make your own cloche by cutting the bottom from a plastic milk jug. The covering retains warmth around the basil and prevents light frosts from settling on the foliage.
Dig up garden basil in fall and pot the plants. Trim back the damaged plant foliage and bring the plants indoors for continued winter production. The basil may survive light frost damage and repotting, allowing it to produce one or two more harvests indoors.