I’d start by quoting Bruce McKinney. The voting was close in most states. Sadly, we only have the voting records of the delegations who were sent to state conventions to vote up or down on the Constitution. As to how popular the Constitution was among the “grass roots” supporters, we can only guess.
Professor Thomas Pangle, in his audio course (part of the Teaching Company) and former Yale professor of mine, addresses this issue briefly in his course of the Constitutional Debates…
The Professor says that the main reason the pro Constitution (i.e. Federalist) side won is because they had a concrete solution and the other side had no solution to the pressing problems that the country faced.
And there was a problem: the potential threat of one of the European powers invading. The great powers of Europe—Britain, Spain, France, and even Russia—were far from giving up their territorial ambitions in the New World. And there was a vast amount of territory to the west yet to be fought over. The colonists were all too aware that without France intervening on their side, they would’ve lost the revolution.
The Articles of Confederation had little hope of creating an American government strong enough to stand up for itself against any of the great world powers. The Constitution made it possible to raise and command a powerful army and navy that could repel invasion.
Of course, the great power given to the federal government, to tax, to raise armies, to give command to a strong executive, had its obvious downsides. That’s why the vote was close in individual states.
Ultimately, the anti-Federalists lost because they couldn’t propose any better solution. But we owe to them their demand for a Bill of Rights.