One of the things that is important to remember is that publicly, there was far more of a taboo on an illegitimate child, on sex outside of marriage, etc., but these things happened quite frequently. People just didn't discuss them openly and they also were more likely to feel that there were real consequences in the form of damnation, etc., that would be applied to those who flaunted the religious and moral rules of the time.
But Hawthorne and other authors of the time were interested in challenging some of these mores and some of these taboos. So bringing out the story of Hester Prynne and the reactions of the town and the hypocrisy of many of the so called leaders is an important commentary on the social fabric of the time and challenging some aspects of it.
One way to answer this question might simply be to consider the plot of the novel itself. How does Hawthorne portray the Puritan society? What values do they privilege and what do they reject? Hawthorne begins the novel by describing bearded men in "sad-colored" garments and "gray" hats and describes the colony as a place where there was the immediate necessity to build both a cemetary and a prison. Many values of the society are suggested from this beginning alone.