I'll upload a savegame played with default rules where AI already have lots of helicopters built, in case it helps.
This sounds like mismatch in the requirements for the AI to build the unit, and in when AI will use the unit. In short; it should not build units that it won't be able to use.
I noticed it testing v3.0.6.
When I edit the default civ2civ3 ruleset to remove the cargo capability from the Helicopters, the AI no longer move them: If I start a game where I removed the cargo capability, the AI will still build Helicopters, but will keep them all inside the cities. As soon as I edit the ruleset to enable the cargo again, the AI starts using them to attack the enemy (even when the AI never uses them to carry units). If I disable the cargo capability again, all the AI helicopters remain in place, without moving, until they die due to HP loss per turn.
I discovered this odd behaviour because I was editing the ruleset to remove the fuel from every air unit (else AI never build them), and to replace it by hp_loss_pct like the helicopters. When I do that, the AI loves to build planes and bombers, hundreds of them, but they never use them to move or attack. If I give them the capability to carry missiles, the AI still won't move them. It is only when the cargo is "Land" that the AI start using them to move and attack.
It is weird, I guess there is a bug somewhere, but maybe it is just the way the AI uses cargo units compared to non-cargo units. In that case, I'd request to allow the cargo = "Missile" to behave the same way, because I like how AI uses these air units (with hp_loss_pct = 10, and cargo = "Land") .