The setting of "The Minister's Black Veil" is the town of Milford, somewhere in New England, which we know from a footnote in which Hawthorne says that "another clergyman of New England" made himself famous for adopting the same "eccentricity" as Reverend Hooper. Given the fact that Hawthorne spend much of his life in Salem, Massachusetts, it is reasonable to place Milford in or near Massachusetts.
Although the fact that the story takes place in "Milford" is of little consequence, the fact that the setting is in New England is very important. In towns like Milford, usually small and agricultural based, the minister is usually the leading citizen, looked up to by everyone, including civil authorities, for moral guidance, religious instruction, and comfort in times of distress. Anything that alters the nature of the town's minister will also affect the town either negatively or positively, so the setting itself is an important element in the overall story even though the town's name is relatively unimportant.
Yes, the setting of the story is identified in the first line of the text: "The sexton stood in the porch of Milford meeting-house, pulling busily at the bell-rope." Milford is a town in Massachusetts, a fact that can be confirmed with a quick google search, but we could also make a fairly educated guess to this effect because of the clues Hawthorne gives us regarding the time period in which the story is set. First, Puritans referred to their places of worship as meetinghouses because they preferred to differentiate their worship from that of Catholics. The Puritans loathed Catholicism and what they viewed as the corruption in that Church; Puritan meetinghouses lacked ornament and the accoutrements associated with Catholic churches. Further, some members of the community are given the title, Goodman, which was essentially the Puritan "Mister." Both the reference to the meetinghouse as well as to Goodman Gray help to identify this community as a Puritan one which means that the story likely takes place sometime between 1640 and 1690. Since the Puritans began the Massachusetts Bay Colony, identifying this group as Puritan helps to confirm the setting's location as well as time.