So I ran into this problem this morning on the Mac client. I was able to find a lot of threads about this problem with the PC client, but nothing for the MacOS client. The PC solution involves resetting IE config or something, but here is how to fix the problem on a Mac:

Open Finder or a Terminal and navigate to here and delete everything in the folder:

~/Library/Application Support/FINAL FANTASY XIV ONLINE/Bottles/published_Final_Fantasy/drive_c/users/crossover/Cookies

In Finder, you can select Go to Folder (Shft-Cmd-G) from the Go menu and paste in the path. You can back the files up to another location first if you want, but it really doesn't matter. After deleting the old cookies, the launcher will work properly again.

Hope this helps someone else out at some point, as this seems to a fairly common and recurring problem.