I would think most Americans don't know who Winthrop is. Personally, I did not until I started researching his life to answer a question at eNotes. However, I found that there are many who do: Ronald Regan quoted Winthrop (who, ironically, was not a supporter of democracy). Sen. John Kerry is related to him. And historians who are familiar with the Puritan settlements in the Massachusetts area, as well as the Massachusetts Bay Company will probably know Winthrop.
(I find it interesting that such a religious man was also so involved in capitalism through trade. In today's society, we don't often hear of the pairing of money and religion. Though Puritans kept Charles I lavish lifestyle afloat because they worked so hard...)
People who know the Bible will be familiar with Winthrop's allusion to Christ's concept of being like a city on the hill, which cannot be missed. Research seems to indicate that Winthrop didn't have much of an impact with this piece when he delivered it, but as humans are creatures who often study where they have been to find where they are going (i.e., historians, philosophers, etc.), his work still speaks to people today. If his writing did not change many hearts or minds then, it certainly affected the way he lived and the example he set, which would have had an impact on those around him, within his society and his family.
So even though people may not be able to pinpoint the concept directly to Winthrop, many will recognize and share his vision, regardless of in what manner this information came to them. I believe, too, that many Americans, especially those with the vision of our founding fathers, still have a sense of being a leader in a global way. We often miss the mark, but that doesn't mean the sentiment—and the desire—is not there.
Most Americans would have a very hard time identifying who John Winthrop is. Therefore, you can say that he did not have any impact on the US at all.
However, I remember that you are doing that project on the idea of the "city on a hill." If that is the context in which you are asking, Winthrop had an important impact (or at least the idea that he made famous has). The idea that the US is supposed to be a "city on a hill" has affected the country, especially in its foreign policy.
The clearest example can be seen in how the US has long tried to spread democracy. The US has felt that the world would be better off if all countries followed its example and so it tries to encourage other countries to adopts its forms of government. This sort of attitude might be traced back to Winthrop's idea.