On which side?
Seriously, the war devastated New England shipping and commercial interests. Why SHOULD they support the war?
This is pretty much the only answer you're going to get, New England didn't get
anything out of the war except for an absolutely devastated economy. I assume you're familiar with the Hartford Convention? That was basically the New England states getting together to agree on an ultimatum to make to DC because the economy was so devastated by the war. An ultimatum which got laughed off the floor of Congress because the war had just ended at that point, but nevermind that.
Basically, there is absolutely no reason that New England has to support the war other than nationalism, which wasn't very strong at the time. I mean, most people identified as a citizen of their state rather than as Americans.
EDIT: I'm going to tell you what would happen if the NEw ENgland states contributed fully to the war effort anyway, even though I think that New England supporting the war as it was OTL is pretty implausible. New England made up a much larger portion of the US population than it does now, and it was home to a large portion of AMerican industry as well, so if New England gets behind the war completely then I could see the US being much more successful in the war, maybe so much so that modern historians would consider the war an American victory. I also feel that the British offensive in the Chesapeake would have been much less successful, possibly without the burning of Washington occurring and Baltimore not happening as a result, which would lead to some interesting consequences since Baltimore is where the Star Spangled Banner was written, so we'd definitely have a different anthem. I'm not sure how the American offensive into Canada would go, though, I could see the US having much more success in taking Montreal, but I feel as though Britain would be able to retake it.