When it comes to the Puritans, crimes were typically punished harshly. In this theocratic society, most sins were also crimes; so, when crimes were committed, they were dealt with unequivocally because the ramifications were eternal. Sin must be eradicated from the entire town as well as in individuals. In The Scarlet Letter, Hester's sin of adultery, for example, actually merited, according to Puritan law and practice, death.
Your question refers, I assume, to chapter two of the novel. The narrator is describing thescene on the morning Hester Prynne was to be released from prison long enough to mount the scaffold for her three hours of public ignominy. He describes the somber and dour atmosphere of the day, explaining that it might have been many things which would cause such a scene of solemnity:
It might be that a sluggish bond-servant, or an undutiful child, whom his parents had given over to the civil authority, was to be corrected at the whipping-post. It might be, that an Antinomian, a Quaker, or other heterodox religionist was to be scourged out of the town, or an idle or vagrant Indian, whom the white man's firewater had made riotous about the streets, was to be driven with stripes into the shadow of the forest. It might be, too, that a witch, like old Mistress Hibbins, the bitter-tempered widow of the magistrate, was to die upon the gallows.
It might have been the "crimes" of a lazy servant or an unruly child of a religious heretic or a drunk Indian or a witch.
In Puritan England (and then again in Puritan "New England"), different crimes warranted different punishments. Puritans were religiously zealous and were especially watchful of anyone who did not follow the laws of their community.
Of course, in The Scarlet Letter, Hester is "convicted" of adultery, and she is not only shunned within the community, but is forced to wear the scarlet "A" on her dress. This was one form of punishment.
For crimes such as witchcraft, treason and murder, those convicted would be put to death. Anyone who has read the play by Arthur Miller called The Crucible will know that in America witches were never burned. They were hung or pressed to death. (There were, of course, "tests" the Puritans used to check for guilt regarding witchcraft, which often resulted in death. For example, "dunking" the accused in water was one such test. If she—or he—floated, she was a witch; if she sank, she was innocent—and dead.)
Drinking, failing to attend church, being vain (this would have applied to women), etc., were crimes punished by whipping, prison, fines, or being placed in the stocks. (While locked in stocks, often food would be thrown at the guilty party.)
In "Bilboes, Brands, and Branks: Colonial Crimes and Punishments," James A. Cox reports:
In l668 in Salem, Massachusetts, John Smith and the wife of John Kitchin were fined "for frequent absenting themselves from the public worship of God on the Lord's days." In l682 in Maine it cost Andrew Searle five shillings merely for "wandering from place to place" instead of "frequenting the publique worship of god."
And woe to the man who profaned the Sabbath "by lewd and unseemly behavior," the crime of a Boston seafaring man, one Captain Kemble. He made the mistake of publicly kissing his wife on returning home on a Sunday after three years at sea, a transgression that earned him several hours of public humiliation in the stocks.