Mine Layers are very similar to Harpooners in a lot of ways. They aren't as tough as the anchormen and they throw projectiles. Their biggest difference are the projectiles themselves, which take the form of mines (similar in ship to those outside of Tiny Tina's but much smaller).
The mines make the Mine Layer much more dangerous than the Harpooner. You still want to dodge the mines that are thrown at you but you need to deal with this enemy quickly because the longer they stay alive, the more mines they will throw. Every mine that hits the ground will stay in place for a certain period of time or until you set it off. Basically, you will start running out of places to dodge to very quickly if you aren't careful.
Because of this danger coupled with their low health (relatively speaking), these should be very close to the top, if not the top, of your kill list when they appear. Thankfully, they only really appear during the mission "Let There Be Light" towards the end of the DLC.