I was just going to comment on UnHoly One's great solution, butI went about Flynt a tad differently and I wanted to ensure people can read his solution and mine (which would be hidden in comments if I just commented).
Once you get off the elevator, there are a set of stairs that lead to the platform you are standing on (on your right side as you face the incoming bandits). All the mobs will try to use those stairs, so you want to stand right there at the top of the stairs as this is a strategic bottleneck.
If you still have the SPR100 STEEL MATADOR shotgun you got from the quest in Old Haven, then use it here. It has a fast rate of fire, you can hit more than one mob, it one-shots some of the bandits and it holds a whopping 12 rounds!
If you don't have the shotgun, get your next best automatic weapon, but keep in mind that a larger magazine outweighs damage here if you only have to give up 5-10 points of damage as you need to keep your reload time to a minimum.
Go for head shots with your automatic weapon, especially on the Badass Psychos. You'll have to judge for yourselves whether or not you should go for head shots with the shotgun. If thre are groups of bandits near you, the knock back of the shotgun will likely outstrip a head shot in effectiveness.
Equip the Transfusion Grenade Mod (health drain) and use them if your shield depletes. As usual on groups of enemies, don't wait for your HP to go down to toss a grenade. If you don't have this mod, then as Scooter says, "Get you one!" I prefer the Corrosive Bouncing Bettie mod (or any Bouncing Bettie mod) if I don't have Transfusion.
Once you finish the bandits off, the Badassandthensome Bruisers come out to play. One ranged and one melee (the ranged one is the one firing his weapon at you). I think the names are Hans and Franz (a reference to an old SNL sketch with Dana Carvey and Kevin Nealon). So don't be a girlie man and run, you should crush them like grapes with no skin and squeeze them like a tube of toothpaste.
Okay, Hans and Franz catchphrases aside, the melee bruiser didn't come for me until I shot him. So I concentrated on the ranged bruiser while still standing on the same platform I fought the bandits on. I used a TMP8.3 incendiary machine pistol, but any incendiary or corrosive weapon will work fine so long as it's not a shotgun. Remember to take cover when reloading and when your shields are down.
When you shoot the melee bruiser, he will come for you. But he uses the stairs just like the bandits earlier. There is a spot in the middle of the platform where you can drop down, so keep firing on the melee bruiser until he gets up the stairs and turns towards you. Then jump down, turn around and shoot him. Sirens can take advantage of Phasewalk here if you need to get away. Use the same weapon as on the ranged bruiser.
So I killed the ranged bruiser first right from the platform I was standing on once the enemies started spawning
Flynt will stay on the top level for a bit after you dispose of everything on the ground and he will fire at you. Be sure to take cover when he does as he fires off 6 shots consecutively. At least two more bandits spawn on the top level as well. Try to take them out with a sniper rifle before Flynt gets down to you. You can take cover and fire your weapon inside a small shed on the left (as you face Flynt's starting area).
Once Flynt is on the ground, be sure to reequip your incendiary or corrosive weapon. He takes a while to reload so pop him when he's reloading and try to entice him to fire his weapon. Once he fires, take cover and then starting shooting him again. Repeat until dead.
There is a ton of cover in this area and you can easily make him follow you around the cover.