I've been conceptualising a story involving just this sort of technology for a while now, and I've had a number of ideas for the weapons that could be used.
One of the simplest ideas is wooden cannon. This might sound insane, but is perfectly plausible - a cannon with a thick, wooden barrel. Functioning in the same manner as a normal naval gun, but many times lighter. The main disadvantage of these guns is that their lifespan is greatly diminished compared with their metal counterparts, and they are more prone to failure after regular use. This won't be a problem if your airships return to a safe port for resupply on a regular basis. There need be no waste either, as the wood from used/expired guns could be repurposed for other means afterwards. Another advantage of wooden guns is that a crew used to naval warfare would already be familiar with their operation.
In terms of the weight of their shot (cannonballs etc.) the Chinese used a projectile called bo-hiya - a large flaming arrow that was fired from a conventional cannon. There are plenty of options that don't require hurling balls of metal at each other.
Ballistic trajectory weapons such as catapults or ballista are another option. Much safer than explosive weapons (fire is bad on a naval ship; on an airship it is many times worse!) and able to fire shells, large bolts, or even just piles of nails, they are extremely adaptable. The disadvantage here though is that their slow trajectory limits their range and accuracy, and mounting them under your gasbag with a clear line-of-sight is difficult.
Rockets were invented by the Chinese centuries ago, and are another viable option. Requiring virtually no heavy firing apparatus (a simple tripod is just about it), these can travel great distances at reasonable accuracy, and inflict massive damage on opponents. However they require competent gunners to calculate the correct fuse lengths.
Gravity bombs are very simply large objects that you can drop on to your enemy. Imagine the damage a tree trunk would do, dropped vertically from a great height - it would punch a hole not just through your enemy's gas bag, but probably straight through their deck as well! Whoever has the greater height will undoubtedly have the advantage in a battle. Explosive shells and grenades could also be dropped onto enemy ships.