I've played the PC one for a few years and I've seen the 360 one in person.
Though I haven't seen/played the PS2 version, I've heard the game is played in a low resolution. Meaning the graphics are the worst. I wouldn't advise this version.
The 360 version would be the most expensive obviously, since you have to pay Xbox Live subscription and PlayOnline subscription. Other than that, the game is very close to the PC version. A keyboard is recommended though.
PC version is hands down the best one. Can play in the highest resolution compared to any other versions, and also loads the fastest. The game does not need the mouse at all, the entire game can be played on keyboard or controller. PC one is also moddable, like most PC games, so that could be a bonus.
All in all, they're all the same game. The only difference is graphics and convenience. PC wins in both, so I recommend that.
Aaaah... a new FFXI player? This is a sight that I haven't seen in a while...
Anyways, I currently play on the PC, and I would say it would be the best due to convenience (alt-tab when looking for party, or browse the web to look up info), and budget wise. With the X-box version, even though you don't need xbox live gold, you pretty much still need to spend money on one of those Xbox keyboards if you even want to function properly in game.
360 graphics are excellent, but if your pc's graphics card is awesome, you can tweak the pc version's graphics to look even better.
As for the PS2... let's just not even talk about it.
Overall, it's the exact same game, and so it just boils down on two things: Your budget, and how good you want the graphics to look.
If you have any questions regarding the game... feel free to PM me or something. I'm on Titan server.
I've played FFXI since it came out like 7(?) years ago and still do.
I even met my fiancee on there!
Anyway, I've played all three versions.
PS2: Laggy, 640x480 resolution so the menu and chatbox take up a good portion of the screen. Poor graphics. But it does the job.
XBOX360: Looks the best of all three. The graphics were updated a little for this version.
PC: Obviously the best, but not because of the graphics. With the PC version, you have access to all sorts of plugins and hacking tools (good (like mini-maps) and bad (like teleporting around)). You can edit the interface, colors, even change the models of whatever you want.
Not only that, but you can change the resolution of the onscreen elements so make the chat box and menus smaller.
The best thing is get the PC version, then see if you can get a used copy of the 360 version since once you make the account on the PC, you can access on any version.
Little trivia for you, the reason that there are some limitations in the game, such as limits on the amount of items you can carry, is because of the PS2. The PS2's memory is limited and can't handle as much as a PC or 360 would, however, you can't really give the advantage to some consoles and not the other. It wouldn't be fair.
I played the game early on and was frustrated by PlayOnline on the PC(not sure if anybody else hated it then) so I haven't played since then but if it's true about 360 only having to pay for Xbox Live then I would say go for the 360 version.