You don't even need to download a separate app if you play FFXIV via Steam; Steam can transfer it over itself. Quick guide:

1) Go to "Settings" under "Steam" in the Steam app. Select "Downloads" and click on "Steam Library Folders"; this will show you the current folders you have available for placing Steam games. Click "Add Library Folder" and then just use the "New Folder" button to make a new folder in your SSD; label it whatever you want.
2) Close out of those menus, and go to FFXIV in your Steam Library. Right-click it and select "Properties." Navigate to the "Local Files" tab and click "Move Install Folder." From the dropdown menu, select the new folder you made on the SSD and then click "Move Folder." Steam will automatically move the entire game over to the SSD.
3) Since I've had issues with this on occasion, once it's done moving it over, click "Verify Integrity of Game Files" in the same tab, just to make sure everything got moved over properly.

And there you go - FFXIV should now be installed on your SSD. Note that I don't use FFXIV through Steam, but I've moved other games in this manner before, so it definitely works, and I do have FFXIV installed on one of my SSDs - the load times are way better than on a hard drive. Enjoy the better loading times!