2012 CNN ORC International poll

Of the over 1,000 polled, 13% said that an American royal family would be a good thing for this country. 85% said it would be a bad thing, with only 1% saying they had no opinion.

What's also interesting is the trend showing a settling increase in the number of people who support an American monarchy, compared to decades ago.

1950: 3%
1999: 12%
2002: 9%

To put things in perspective, keep in mind that about 11% of Americans find a Communist system to be "morally superior" to the current American system. This means a tiny bit more Americans support a monarchy than they do Communism.