One of the ammo types is the heatsink round, though, which puts out fire automatically, and should be in ever gunners arsenal. I do tend to put a flamethrower on most of my ships, but it's primarily an annoyance weapon - it gives them one more thing to worry about, but it isn't the cause of any of their problems, just gets in the way of fixing them. They need to sit in front of it for a LONG time before a thrower brings down their weapons, and most of the flame ships tend to be pretty fragile.
The flamethrower is definitely the easiest to use though - although you need to be close. So it tends to dominate the lower tiers, but isn't nearly as useful against a ship with a good captain and a halfway decent gunner, since the flamethrowers will be knocked down before they get into range. They might come back up long enough to do a bit of damage before getting knocked down again, but that means someone is ignoring the other systems going down so it's still a win.
So basically:
Gunners should always bring heatsink ammo, it makes the gun immune to fire damage.
Captains need to control the range of their ship, and check the enemy weaponry before going out. Don't close with flame-ships.
The crew need to take down the enemy weapons sooner rather than later, and Captains need to provide them with the right guns and opportunities to do so.