Yes there are airships in this game. The thing is that we are playing in under developed cities. I mean, the video for the opening with the airships fighting the dragons is actually happening now. You can even see the giant ship crashed in a zone north of Ul'dah.
All I know of is that there are platforms around the rank 10 camps for airships, but we cannot use them yet.
Actually, the battle we saw during the opening cinematic happened ~10 years ago, as the Midlander in the movie was experiencing a scene from the past - the Echo ability that all adventurers seem to possess. If you also notice during the cinematic, when the ship crashes, avatars/primals go flying out of it, which makes sense as well, since it was 10 years ago that the Primals first appeared and the beast tribes started causing trouble.
It is nigh on ten years since the fearsome beings known as the primals first appeared in these lands. Ten years since they were summoned forth by the destructive ambition of the beast tribes. Ten years, and still those tribes, drunk on the power granted them by the primals, continue to wreak havoc upon the free peoples of Eorzea.
10 years ago is also when the Garlean Empire ceased their advance into Eorzea after taking Ala Mhigo. We're not told why they stopped, but I'm guessing the assault of one of their giant battleships from the dragon (god/Bahamut?) was the likely cause. Edited, Nov 2nd 2010 1:53am by Serielley