Apparently Xbox users are not suppose to worry about stuff like that (or maybe I'm missing something). I don't see any way to check the game version or to verify if the patch was downloaded. If you are not getting a prompt, though, you more than likely already have the update.
You can check the size and date for the game in the System Settings, Memory, Hard Dive, Games, D2:ED section; depending on whether you have the game installed or not, and when that was done, that might confirm the patch was downloaded. In my case I installed the game right away, so the 5.5GB size doesn't determine anything, but the date is later (bought it at release), presumably matching when I downloaded the patch.
I remember installing the patch, since it took me awhile to get around to buying an adapter to reach across the room with an ethernet cable (the wireless adapter being way too expensive), and once I had that it took me 10 minutes to get online (most of which was trying to remember the hotmail address I set up to download the Windows 7 beta) and 30 seconds to get the patch.