It really depends on where I'm fishing.
Most of the time, I'll use a stout parabolic rod--something like an Arra 126 or 1205--and a moderate drag. Even in some places in the Canal where I know it's fairly clean downcurrent. If I'm using braid, and plugs with treble hooks, I rarely try to horse a fish.
In some places, you have to put the boots to a fish. There's just too many things in the way that will cost you if the fish can use them. That's where the All Star 1209, the Saltiga with 12 lbs of drag and 65# braid get used. With jigs. If the jig hook is buried, you can't pull hard enough to damage the hook. If it isn't, they fish will throw it eventually--even when it's running. It's only with plugs and trebles that the hook is your weakest link. Unless you're foolish enough to be fishing with a cheap-assed snap like a Duolock or old fashioned safety pin snap.
I still like to use mono for plugging and for eels. It really depends on the lure style. I have problems with darters, especially, when it comes to pulling trebles loose with braid. And you don't need the sensitivity of braid when fishing eels. It might even be a disadvantage--the fish might feel you as well as you can feel it

And the repeated casts with no-stretch line seem to result in the hook tearing loose on the cast faster.