To make fresh herbs last longer, just keep them in a small cup or jar of water in the fridge. Just like flowers! I use canning jars without the lid part because I always have those around, but you can use any small containers you have handy. This makes them last at least a week or two, depending on the herb. Definitely a lot longer than if you leave them out or in those little clamshell packages from the store! Plus I think they look pretty sitting in the fridge.
Looks nice, but sweet basil turns brown at 40 degrees, a temp higher than a good refrigerator maintains. Muy muerto. Just put them in a glass of water on the counter, or use them in a flower arrangement till you’re ready to chop.