The official Civilization Youtube channel actually has a video on this. The key take away here is that Barbarians now send out scouts to seek out civilizations. Once the scout finds a civilization it runs back to it's Barbarian Encampment, which will then start to spawn military units to attack the discovered civilization. Until the scout reports back the Barbarian Encampment won't spawn any military units, with the exception of one garrisoned unit (which in my experience was always a spearmen).
So, with that said, the best way to deal with Barbarians is to either never let the scouts report back to their encampment to begin with, or to just follow them back to their base so you can wipe out the encampment before it has a chance to spawn too many units. Since scouts are pretty fast it's likely that you'll be doing more of the latter than the former, but if you keep a few military units (including Archers) near your borders than you might be able to wipe out the scouts before they report back. You'll still want to eventually track down the encampment and wipe it out, though.
Army wise you shouldn't need anymore than four units (two ranged, two melee) to deal with any encampment you come across, but should be able to just get away with two units if you make sure to hunt any encampment down the moment a scout appears.
Finally, and this final part is just speculation for now, but I think a scout only counts as discovered your civilization if it comes within line of sight of your borders. If true, then you could just keep a few military units stationed outside of your borders to scare off any scouts before they see your borders. If they don't discover you then they'll never retreat back to their encampment to report your existence in the first place, so you don't need to worry. Again, though, this is just speculation for now.