I can see this game taking two general paths.
1. Trying to avoid barbarian troubles and hoping they will be gone as soon as possible. In a regular game the barbarians stop being a threat after most of the world has been colonized. We'd just play our own game and try to win as soon as possible. This would involve staying at war and relying on the AI to kill barbarians.
OR
2. Trying to leverage the barbarian advantages as much as possible. Especially if they make trading difficult. Two low level techs in the early game, might turn into more useful techs in the later game. A few axes can do some damage, but what if we equip them with maces and grenadiers.

This means protecting the barbarians, with extra defenses and workers to help growth, even culture to grow borders. The barbs can keep giving us free technologies. Two low level techs in the early game, might turn into more and more useful techs in the later game. It basically acts like the internet filling gaps in the tech tree.
My preference goes out to option 2, especially because winning normally is difficult with non-trading AI. I also feel barbarian bonuses can be a great advantage if they get out of the classical age.
SOME IDEAS ABOUT HELPING THE BARBARIANS:
1. Focus on military techs to give barbarians better units than the AI, Axes, macemen and grenadiers.
2. Signing peace to increase number of barbarian units. Then declaring war in an area where the AI is weak and barbarians are strong
3. Supporting barbarian cities. When the barbarians get/found a city we should send defense units and workers over to assist growth, maybe even a missionary for culture.
4. Getting the barbarians more cities. Send an attack force to the AI and weaken the defenses to one (preferably wounded) unit. The timing needs to be just right so that the barbs can take over the city before defenses are restored. This also saves us maintenance and gives barbarians a stronger base to build units.