One of the best-known Miyazaki films, Howl's Moving Castle features a giant cobbled-together chicken-legged ambulatory castle, and also (briefly) has some airships. What distinguishes these airships from many others is that they don't have gas bags - they're held aloft by some unspecified force.
Girl Genius is a long-running graphic novel about mad scientists (called Sparks) and their creations and power struggles. I want there to be a Girl Genius computer game, and Airships is sort of an unofficial and partial attempt at it.
The whole mad science thing is a big influence: creating giant ill-advised war machines for your boundless ambitions, equipping them with dubious new weaponry, attempting total aerial conquest. Wait - you didn't think you were the good guys, did you? Have you seen the airship names?
Apart from the obvious influence of "2D game about laying things out on a grid and having units run around", DF was also influential in my decision to not allow direct unit control. I know that that's a rather bold decision, but if you could order around individual airsailors, the game would descend into a madness of micromanagement.
Early on, I described Airships as "like FTL, but with ship design and no direct crew control". The actual experience of play is pretty different, but mechanics like having to put out fires were definitely inspired by FTL's frantic combat.
Cortex Command is basically a fancy remake of Worms, which is an absolutely worthwhile endeavour. With Airships, it's inspired the side-on view and destructible terrain, and the cheerful mayhem of most Cortex Command matches is something I want to aim for.
Beyond this, there are a number of honourable mentions: HG Wells' Cavorite is the inspiration for Suspendium, about which a detailed post will soon follow. Master of Orion II, one of my all-time favourites, is probably what turned me on to ship design as a game mechanic.
In the end, Airships is developing into its own thing, and it's a thing I very much like - but the more inspiration gets added into the pot, the more fun the end result will be. What I want to end up with is a combination of calculating ship design and chaotic warfare: picture a burning airship against the smoky sky, with a crewman jumping out, blade at the ready.