As far as I've tried so far, the fastest is with two shotguns, stowed reloading, under the Banshee, jumping into the air as close as possible to his belly (the low gravity helps here) and shooting from close range. Even when he throws the mines, if you're in a jump when they explode down on the ground, they don't do any damage.
That's how it works with Spire sites, but not in settlements (with ax reactivations) because Revenants keep appearing there, so you can't fight Banshee one-on-one. I was killing him there with the Scorpion, from a distance.