The only major disease of any significance that affects mint is mint rust, it's a fungus called puccinia mentha. You can tell if your plants have this fungus by looking at the underside of the leaves. It identifies itself by covering the leaves in rusty spots which can be orange, yellow or brown, most people expect the disease to look rust coloured but that's not the case, although they can be rust coloured..
The only organic way to get rid of this disease is to dig up the plant and get rid of it, if it is mint rust, throw away the affected plant and compost, scrub the pot real well using hot soapy water, and try again with fresh compost and a new plant. You can't just cut off the leaves that show signs of disease, it can live in the soil still and the fungal spores can spread in the wind and remain in the soil for at least three years if not longer.
Lemon balm mint is a new one on me, I'm wondering if you're getting confused because lemon balm is a member of the mint family? Keep an eye on the lemon balm as it to can be prone to rust as its a disease that can effect the entire mint (labiate) family. Hope that helps?