How did the Puritans view sin, guilt, crime, and adultery?

The Puritans believed that their church had entered into a covenant with God.  In this covenant, they pledged to act correctly in return for God's special favor (on Earth).

Because of this, they saw sin, crime, and adultery as violations of their covenant with God (and the covenants they made with each other to act in accordance with God's will).  These things had to be punished by the leaders of the church community so that God would not punish the society as a whole.

The Puritans did not, however, see sin as something that would cause people to be damned.  People were damned or saved based on predestination and nothing they could do would change their fate.