Without looking at these plants closely with a microscope the the best I can say is this is a fungal infection, I suspect its Fungal Blight based on the lack of yellowing around these spots on the top of the leaves.
In terms of treatments well.. the prognosis is grim, unfortunately. You might try an over the counter fungal treatment or some natural/organic remedies you might try:
- Apple cider vinegar spray: 2 tablespoons/gal
- Baking soda spray: Mix one tablespoon of soda with two and a half tablespoons of vegetable oil and add to a gallon of water
On top of this your going to want to throughly rake the patch and clear the bases of these plants of debris. Prune (heavily) to increase air flow and slow down your watering schedule.
As I said the prognosis is grim. The "infection" has seemed to spread throughly. If you have sections of the patch still uninfected than I suggest you simply remove the infected sections entirely. Be sure to sanitize your gardening equipment before and after working with your mint, you woulnd't want to infect your tomatoes or what-have-you. These fungal infections can and will spread.