Keeping you fresh mint fresh is actually quite easy to do. This is a guide about storing fresh mint.
How do I keep the mint leaves fresh?
Wrap the mint in a moistened paper towel. Place the towel in a plastic bag and seal it, and then place the bag in the vegetable drawer of your fridge. Good luck.
I keep mine wrapped in the store veggie bag on the top shelf of my fridge (I do this with all fresh herbs) and they stay fresh for about a week and a half. Plus I don't forget I have them storing them that way because they're not hidden away ;-)
When it looks like they might be going bad and haven't used them all yet I place them in a small freezer bag and keep them on the door compartment of my freezer because they'll still be good as a cooking ingredient :-)