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In British English you hear both. I'd use the second, but the first works, though purists might want to put a hyphen: new-year resolution. I'm not sure I like the hyphen and, anyway, prefer to use the possessive as the resolution does belong to the new year. In fact, given that most don't last beyond the first couple of days of the year, there is a very strong case for the possessive here.
"New Year" or "New Year's" resolution 
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