How did people of Nathaniel Hawthorne's time feel about guilt, sin, crime, and adultery?

Your question can be answered in two ways.  If you are referring to Hawthorne's time period of the mid-1800s, then Hawthorne's contemporaries had diverse views of the human condition, especially when it regards the idea of sin.  For example, Hawthorne was a member of a group of writers called the Dark Romantics (Poe and Melville are also categorized in this group).  The Dark Romantics believed that man's nature is basically evil and that he is prone to sin and displease God.  In contrast, the Optimists from this era (Emerson and Thoreau) believed that man is inherently good and that he can reach perfection or at least attain high ideals.

I think that your question might, however, be in regards to the time setting of Hawthorne's The Scarlet Letter.  If that is the case, then the Puritans (Hester's contemporaries in Boston) maintained strict standards and exacted harsh punishments upon those who "sinned" or committed crimes. This derives partly from the fact that the Puritans see God as a wrathful being who is waiting to punish those who displease Him.  No Puritan could know for certain if he or she was going to heaven; so it was extremely important for Puritans to live "righteous" lives with the hope that they would be good enough to be "covenanted" (meaning that they would be allowed into heaven). If a Puritan committed a sin or crime (for the Puritans, sins and crime are the same thing), he/she must be dealt with promptly and should feel the burden of guilt and shame.  If the offender did not feel guilt or shame for his sin, then the Puritans believed that that person was not covenanted.

Adultery, for the Puritans, was a sin/crime that could be punishable by death.