Well, you could do it via a single column and field that says 'IsDateComplete'.
If you only have the date field, then you'll need to encode the "incompleteness" in the date format itself, such that if the date is, say, < 1900, it's considered "Incomplete".
Personally, I'd go with an field on the side, that marks it as such. Easier to follow, easier to make decisions on, and allows for any dates.
It goes without saying, perhaps, that you can just create a date from DateTime.MinValue and then set what you "know".
Of course, my approach doesn't allow you to "know" what you don't know. (That is, you don't know that they've set the month). You could perhaps use a date-format specifier to mask that, and store it alongside as well, but it's potentially getting cumbersome.
Anyway, some thoughts for you.