Barbys who attack from water, you can't do much about, in the early or mid-game. Roads and 2-move military units, and tight clusters of cities, is the best defense. Barbys who spawn on land, however, you can eliminate. Notice them cropping up in a certain patch of land repeatedly? Just build a city there. The insurrections will move farther away. Eventually, as your cities cover every desirable patch of land, the Barbarians will only spawn on the North or South Pole, there's a flash of a red unit on snow, and they immediately die of frostbite. Rule of thumb: any place where you see Barbys pop up, that's a good place for you to build a city. Roll a Settler up there with a Cav/Char/Kni, found a 'burg and start another Cav/Char/Kni in your civ's core to move that way and protect the far city.
Every lobe of your landmass which you can populate, decreases the chances that Barbys will arise on your land. Pleasantly, it increases the chance that land Barbys will pester your rivals. Unpleasantly, far separated cities are harder to protect. Your game will be different, but in some games i have found it useful to keep a stalemate on a rival instead of killing their capital, just because the enemy city prevents Barbarians from spawning on that spit of land.
