They can be live trapped. Go with a smaller trap if you have one. If you have one with doors on both ends (Havahart sells some. Not the best trap, but will do), set it and cover the trap with cloth or the such. Leave the ends open and exposed. They are a curious animal and the tunnel effect may lead them to enter out of curiosity. Wire bait above the pan, as an extra incentive, so they have to work it a bit. I chicken head, chunk of liver, or such can be effective. Wiring the bait in place will keep them moving about in the trap and increase the likelihood of tripping the pan. A trap with a door on one end is okay. Set up the same way with both ends exposed.
You can often see trails in the taller grass along streams where they and other smaller animals travel. Set in the trail. or along the bank at the edge of the water. They are low on the food chain themselves, so will not spend much time out of the water where there is not cover. Raptors, foxes, other mink, etc... will take their toll.
Creating a trail out of the water can be effective as well. Stabilize that trap with rocks with one end facing the water, maybe 5 or 6 inches from the water. Slick up a trail (water and mud) to it from the water to the mouth of the trap. If it is grassy, create a trail or a few trails from the other end. You have a good chance of catching other animals as well: weasels, muskrats, skunks, etc... All can be released unharmed if desired.
I'm not sure if mink are really sensitive to wire, but you may want to cover the bottom with some grass or enough dirt to cover the wire, but not hinder the working of the trap. Some will glue a bit of carpet or thick cloth on the pan as well. Not necessary for coons, but some animals are more sensitive to the feel of an unnatural material like metal.
Just a few ideas...