Not sure about exactly what it is. Maybe a fungus which can be caused by not enough light combined with too much water and/or soil that does not drain well.
I moved about 20 organic basil plants in recently. I kept them in very small pots and placed them near a window. I let the soil dry out slightly before I water them. They are really perking up. Hopefully I can keep them going until ready to transplant to bigger pots once it is warmer. Keeping them in small pots helps with drainage it seems because of the root-to-soil ratio. I have some sage that is in a larger pot with a lot of dirt that isn't doing as well inside. The sage that I have inside in a very small pot is doing great. I think it has to do with being able to dry out and not get stagnant.