In a lengthy forum post, producer and director Naoki Yoshida (aka Yoshi-P) outlined what steps have been taken to stabilize FFXIV's game world. Yoshida noted that all Final Fantasy XIV servers now have login restrictions to prevent a flood of logs and re-logs from causing them to crash. Login restrictions are most likely to be implemented during peak times, which Yoshida says are between 9:00 p.m. and midnight. Each server can support a theoretical maximum of 5,000 concurrent players.
Character creation restrictions have also been implemented; Yoshida explained that while the worlds are designed for every player to have multiple characters, they cannot handle all of those characters online at once in the same place (i.e., the starting zones). Until players level up and spread across the world, character restrictions will be a must. The FFXIV team is conducting 24-hour surveillance to ensure changes are made as needed.
Square Enix is in the process of adding new servers to accommodate new players; the hardware is in place but requires further testing before it can go live. According to Yoshida, players have flocked to the game a bit more enthusiastically than expected, with peak concurrency reaching over 218,000 users.
Finally, Yoshida answered concerns regarding AFK players. He explained that Square Enix is aware of the idling problem in Final Fantasy XIV (players are leaving characters idle instead of risking being unable to log in, thus blocking other players from logging in) and is considering solutions but is focused primarily on adding worlds.
[Thanks to Brandon for the tip!]