Thanks for the link! I have all of the Spelljammer stuff, but that's still useful.
Dumb question for the OP, please bear with me.
Are your PC's fighting aboard the the airship, or are they piloting the airship and attempting to shoot down, pursue, or capture the opposing airship?
If they're the equivalent of naval marines, it's just regular D&D combat with an interesting environment. Add rules for jumping/swinging/teleporting/flying between the two ships and maybe pick some special terrain or items to interact with (Spike the guns! Slash the rigging!") and you're done.
If they're flying the ship, it gets ticklish. I recommend you pick a skill for piloting, have the PC's elect a pilot, and then do opposed skill checks against your NPC pilot. The best score loses or gains 10 feet, gets on their enemy's tail, or breaks away. Really bad failures lead to crashes or stalling out. The other PC's can aid the pilot or fire weapons at their enemies: ship or personal.
Star Wars had a system like this.
It's not perfectly realistic, but I've played perfectly realistic 3D combat games before and don't recommend them unless you not only understand vector addition, but you think it's fun.
And really, this is D&D. If your party is composed of a dragon man, a cat woman, an intelligent bush, a magic robot, and a crystal man it seems unfair to say air combat doesn't look like it did in WWII.
Type of airship matters as well.
- Flammable Gas: Risky in a world of dragons and flame or lightning-slinging spellcasters...not that likely, IMHO.
- Heated Gas: Less risky, but still with some interesting challenges...all of which are shared with Flammable Gas airships. A fire spell could cause an altitude jump; other elemental spells could cause a plunge. Summonings of physical creatures could do likewise. Certain attacks could puncture the gasbags...
- Magic: Less risky and has any number of forms- Enslaved Air Elementals, Brooksian "diapson" powered ships with parse tubes, Wellsian/Space:1889 style cavorite steamers & liftwood sailing ships, Spelljammers and so forth.
- Lost Ultra Tech: Least risky, but to some, the least satisfying because it implies your campaign world is Post-Apocalyptic...or at least "Post-Atlantean." Still, it makes for fun- see Brooks' Shanarra books, Corben's Den, or simply listen to The Sword's Fire Lances of the Ancient Hyperzephyrians
On a certain level, it makes sense- raiders might be willing or even forced to use riskier tech if they can't get their hands on the safer stuff. (That is, from what I understand, part of why the Hindenburg was using hydrogen instead of helium.)
To jazz things up, an additional rationale for the raiders using flammable gas is as a rudimentary form of jet travel. Yeah, I know its not realistic, but perhaps they use the flammable gas for both lift AND for a "rocket" propulsion. This would enable them to close on target craft quickly, like pods of orcas chasing down right whales...
Its a risk/reward thing: they risk blowing up (either by accident or by attack) in exchange for the increased likelihood that they can close on and disable their prey before a reasonable defense can be mounted, improving their odds of a successful raid. IOW, mo' problems for mo' money.