MMO-Champion has been doing some Armory scraping and produced some statistics on WoW's new raid finder. Out of a sample of 4.4 million level 85 characters who have been active since patch 4.3 in December, over 30% have completed the Deathwing Raid using the new automated raid finder on easy mode.
As Chaud notes, it's hard to get to the proportion of players who have completed the raid. The percentage of players who have cleared the zone can be no higher than the percentage of characters (i.e. if no one had ever cleared the zone on an alt, which is unlikely). We can almost certainly say that 70+% of players who have a level 85 character that has been active since 4.3 have yet to clear the raid finder.
Likewise, one imagines that many traditional raiding guilds are taking their people through LFR for gear and practice, which would mean that not all of these folks are new to raiding. A part of the long-term test of this system will be how many players beat the raid finder once to see the grand finale and then never return.
Even so, the numbers have to be viewed as a success, as they will only increase over the remaining months until the new expansion. The dungeon finder effect also has a way of amplifying itself - as more people beat the content, groups will gain experience and be better able to triumph even with newbies on board. For better or worse, this feature is assuredly here to stay.