'Then what is Durin's Day?" asked Elrond.
'The first day of the dwarves' New Year,' said Thorin, 'is as all should know the first day of the last moon of Autumn on the threshold of Winter. We still call it Durin's Day when the last moon of Autumn and the sun are in the sky together. But this will not help us much, I fear, for it passes our skill in these days to guess when such a time will come again.'
It appears that Durin's Day in 2011 was in late November, toward the threshold of Winter.
Winter itself starts on December 22nd, 2011 at 0530 UTC, with the next new moon on the 24th. That means that the "last moon of Autumn" is about a month earlier, on November 25th at 0110 EST (for my location).
For Saturday, November 26th, the sun sets (for my location) at 1659 (4:59 pm) and the moon sets at 1829 (6:29 pm). The moon on that day and time is 3% illuminated (waxing crescent). If it had been clear, it would have been a great time to observe Durin's Day.
The folks at Moon Watch seem to agree.
