What size pot is it in? When mint is root-bound it will lose leaves.

F. DeBaggio

I grow spearmint in a large container. In my experience, mint does best in full sun. I have had problems with leaves turning brown at the base of the plant. It happens when the mint gets too thick or crowded. I believe it happens because the bottom of the plant doesn't get sun. I found that thinning or cutting back some of the mint to the soil line helps. It is also helpful to pot up to a larger pot if it is crowded.

Spearmint is very susceptible to a form of fungus called Rust, which causes the leaves to turn brown. You'll need to get rid of all those fungal spores.

MOLASSES SPRAY FOR RUSTS:
1 cup sulphured molasses
1 cup powdered milk (optional)
4 litres water
1 cup seaweed powder (optional)
1 cup rock powder (optional)

If using only molasses, stir into water and use. If using all ingredients, mix molasses and powders and make into a paste. Wrap 1 cup of the paste in pantyhose. Put in the water, and let sit for 2-4 hours. Strain and spray.

Or you can use a general fungicide like Garlic spray or this one:

FUNGICIDE/POWDERY MILDEW SPRAY:
4 litres water
3 tablespoons bicarbonate of soda
1 tablespoon household bleach
1 teaspoon dishwashing detergent

Snip and remove leaves that are worst affected. Mix ingredients with water. Spray remaining leaves top and undersides. Apply a heavier dose on leaves that have signs of infection and only lightly on unaffected leaves as bleach can actually harm and discolor the leaves.

Remove all those brown leaves plus any that are showing signs of browning, and burn them. You'll need to remove the plant from the existing soil/potting mix, and wash it, roots and all in warm soapy water to which has been added a little bleach. You'll need to wash the pot in a similar solution. Don't rinse.

There is another method, if your pot isn't plastic. Cut the mint right down to soil level, cover lightly with some dry straw, and set the straw alight. The heat of the fire will kill any spores on the surface of the soil, but it won't kill the plant's roots, and the plant will easily resurrect. Of course, you don't need to leave it all in the pot - it can be removed before setting it on fire.