Airships aren't that effective as war platforms.
They are slow, can be spotted from afar (yes, even at night, unless you choose a moonless and starless night which makes things difficult for navigation), their path is somewhat predictable unless you equip them with noisy engines.
Adopting guerrilla tactics and never having much in any one place that can be targeted from above would be an effective defense. Large movements of troops would be spotted by airships, but that's not a very large advantage against guerrillas.
The enemy could largely ignore the airships and concentrate on the ground troops, which would be (often) forewarned and forearmed, but can still be ambushed and harassed in other ways.
However... Thai lanterns.
Observing that the hot air above a fireplace goes up is enough to make people suspect that hot air rises by the sole fact of being hot; from there to Thai lanterns the road isn't long. Earth Thai lanterns are designed for neither height nor endurance, but both are achievable.
Now, swathes of Thai lanterns going every which way, each trailing a long, light, sticky tail of combustible material (twine soaked in pitch?), could play merry hell with airships filled with flammable hydrogen. Once a tail is attached to the ship, when the lantern plays out the fire will reach the ship's balloon, and wetting the balloon or having someone climb on the outside are both very difficult propositions. It's not easy to set an airship on fire using projectiles, true, but using tinder...
Normally an unencumbered hydrogen balloon could rise higher than a Thai lantern, but a war balloon is far from unencumbered, and it is slow.
Thai lanterns are inexpensive and lots of them can be fired in waves, almost simultaneously, using several possible inexpensive devices.
This guy here has run a lot of experiments with Thai lanterns. Granted, they don't have much lift; an average-sized 40 L balloon develops around 0.08 N of force, and with G being 9.81, that translates in a paltry 1 gram of payload every 5 liters of heated air volume.
Still the same guy showed that a hotter fire and, obviously, a larger balloon allow more powerful lifts (he measured up to 1N, or 100 g of payload, using a 500 °C "flame" and a 270 L balloon).
Since the payload goes up with the volume, everything scales in the same manner. The efficiency goes up with larger volumes though, because heat is lost through the outer surface of the lantern, and that goes up with the radius squared rather than the radius cubed. So, larger lanterns will have the same ascensional speed (if fully loaded) and the same lift ratio, but I expect them to have more endurance.
Twine can weight as little as 40 grams per 100 meters, and pitch or other flammable and sticky substance shouldn't add more than, say, other 40 grams. Length also scales better, since we don't use thicker twine for larger balloons.
By pre-heating the air in the balloon (basically, light a fire, run a tube into the fire and blow hot air into the balloon, while preventing it from taking off), it ought to be possible to achieve a strong initial lift and time the release of the balloon to maximize the chances of collision with the incoming airship, thus needing less twine.
The largest paper lantern ever made had a volume of 1,086 m^3, or about one million liters; using the lowest figure for the lift (5 L for one gram), that gives 200,000 grams - or 200 kilograms - of lift. Granted, that lantern had a radius of five meters or more, so it would be expensive to build and really difficult to launch. But an airship is much more so on both accounts, so it might be worth it to deploy a line of "vertical launchers".
That kind of lift would even allow a suicide archer to fly nearer the airships and cover the remaining distance using fire arrows, greatly increasing the chances of setting fire to an enemy vessel.