Just take care of the zeppelins on the right of the ship yourself and use the songbird to kill the infantry.
The ones on the left though you have to use the bird for them but again use your vigors to distract kill the infantry at this time and shoot down the smaller carriers with the turrets on them yourself.
This isn't some Video Game, this is DARK SOULS. - Souldepriver
As soon as the zeppelin spawns, go for it (you can't attack it before it launches it's patriots unfortunately).
When a rocket barge shows up at the front of the ship, have songbird deal with that (the cooldown is not as long as the zeppelin).
I managed to get finally do it with shock jockey (set traps in the path of the patriots) possession (possess a patriot or rocket launcher enemies) and undertow (to launch low level baddies away when they gather around the ship core thing, and you can get rid of rocket guys much faster by tossing them off the airship).
If you are getting overwhelmed and everyone is attacking the core and gathering around it on the little platform below it, make sure you use songbird to squash them, the cooldown is like 8ish seconds for the stomps.
Weapons I used were shotgun, volley gun, rpg and sniper rifle.
"You stay classy, San Diego." - Ron Burgundy "**** you, San Diego." - Ron Burgundy
Vigor usage is really important in this fight. Possess the Patriots or spam Shock Jockey on them to stop them from hitting the ship. You Bronco or Crows on the infantry (or you could use Shock Jockey if you got the chain lightning boost).
I did this one 4 times before I beat it, when I finally did beat it, I was just spamming vigors not worrying about running out of salt because there is so much salt around.