What do I do for an indoor mint plant with dry leaves and stems which are falling over? Some leaves are brown or black.
From the question, it sounds like your mint plant is too dry. Mint plants in pots are very sensitive to the moisture content of the growing media. They prefer very moist, but not saturated soil while growing. Also, mint plants like lots of light.
In low light conditions they will stretch and become weak, which would cause the falling over. Also, the lower leaves will dry out or rot, depending on the water level in the air and the soil. If this is the case, cut the stems individually back to 4-5 healthy leaves, and move the plant into a well-lit area (preferably outside). If the stem bases are black or soft and wet, your plant is dead.
Mint plants also have a dormant period, and will start dieing back when ready to go dormant. In this case, cut the plant back to the base and move it to a cool dark place and water sparingly. It usually takes only two months before the plant is ready to grow again. You can then move it into a well lit area and start watering. You should also fertilize every other week with the watering, for good growth.
The same thing was happening with my chocolate mint plant. It might not be this easy in your situation but I just had to move it outside, put it in a bigger container, and then water a little more frequently. If your base is brown, your plant is probably dead. However, before you throw it out, check for new growth and the bending of the stem and branches. If your branches bend, your plant is hanging in there.