Like many plants, lemon balm will become dormant towards winter.

Being a member of the mint family, it likes its soil to be constantly moist. This can be a problem when the plant is in a pot, as potting mixes usually become water-repellent when allowed to dry out completely. This means that most of the water just runs down the side of the pot, and isn't absorbed by the potting mix - therefore isn't made available to the plant.

Try immersing the entire pot, plant and all, in a large tub for a few minutes. Leave it until you see no air bubbles rising to the surface. Then remove the pot, and allow the excess water to drain off. This will rehydrate the potting mix.

If it's beginning to get cool where you are, don't worry if your plant is looking a bit ratty. Cut off the ratty bits, keep the soil moist but not wet, and it will bounce back come spring. Like most mints, lemon balm is as tough as old nails. It would be a good idea to repot it in spring, anyway, because, sure as eggs, it will be potbound.

It might be too wet now. Some of the roots probably died from lack of water. Try letting the soil dry out a little between waterings and see if it starts making new growth. You could also prune it back to a couple inches tall to stimulate new growth.