Do not attempt to use Barrage or Artillery to destroy the fence. It will not work.
When possible, find different routes that avoid barbed wire when using Zookas.
If you have to go through the Barbed Wire, consider using Troops other than Zookas.
If there is only a single row of fencing in a select area that must be crossed to carry on an attack, place a Medkit on the Barbed Wire as Troops pass through. Examine the heal per second of your own Medkit before attempting as having a net loss could result in a failed attack.
The real-world Barbed Wire was invented in 1865 by Englishman Richard Newton.
It is the only building that cannot be destroyed.
Troops that come in contact with the Barbed Wire will turn a bluish tint until they cease taking damage from the fence.