If you are using Xfire make sure that disable the in game chat system. For some people (like myself) it caused a crash to desktop, for others it runs fine as is.
If you are not using Xfire make sure that your firewall is configured to allow all access to polboot.exe and POL.exe. ALso (not recommended but a good test) try disabling your firewall to see if that makes a difference.
If this still isn't working its time to check your router's firmware to make sure you have the necessary ports open for FFXI to run. As quoted from the Playonline Support Site:
PlayOnline(including all related software such as FINAL FANTASY XI) may use any of the following ports. If you are using a router, please configure it to allow packets to pass through these ports.
Port numbers that may be used by the PlayOnline server group.
TCP 25.80.110.443 or 50000 - 65535
UDP 50000 - 65535
Port numbers that may be used by terminals such as the PlayStation(R)2.
TCP 1024 - 65535
UDP 50000 - 65535
*Your router may require that you use a PC to configure your network settings. For more information about how to configure your network settings, please refer to the instruction manual that came with your router or contact the manufacturer directly.
*If your Internet service provider (ISP) specifies a DNS server, you may need to configure your settings to allow packets to pass through UDP Port 53 as well. For more information, please refer to the instruction manual for your router or the information provided by your ISP.
If you are still unable to maintain a connection to FFXI all I can suggest is check the simple things that we always overlook, like if you're wireless make sure the signal strength is strong, or if you are wired make sure the physical connection is secure.
Good luck!