How did the time period influence The Scarlet Letter?
I am not sure if you mean the time period Hawthorne wrote "The Scarlet Letter" or the time period in which it was set. I shall try to provide both answers for you.
First, Hawthorne was interested in documents he had come across while reseaching his own ancestors. He discovered that a woman had indeed been branded with a letter "A" as punishment for adultry. Hawthorne became fascinated with the religious hypocrisy of the Puritan era. By the time it reached American shores in the late 1600s, Purtiansim had become a corruption of the original ideals that led to the travesties like the treatment of Hester and eventualy the Salem Witch Trials. Puritanism suffered a severe decline in adherence thereafter.
As for Hawthorne's time, he offers his "Scarlet Letter" partially as a warning to those who would read it in modern times. Crowds can easily become hysterical with little or no basis in reality. For example, on pg 116 (my edition) he writes, "When a multitude atttempts to see with its eyes, it is exceedingly apt to be decieved." Or, later (pg 185), "Prison is in our own minds."
The message, therefore, is that we should not be fooled into thinking the Puritan acts were some sort of abberation. They could, have, and will continue to occur. Just look at the headlines any day of the week...
In addition, the fact that the book is set in the time period in question helps bring forth the point that Hawthorne was raised by a Puritanic father, and Hawthorne chafed under that level of strict parenting. You can see that in his distaste for the people of Boston.
Further, Hawthorne's owns beliefs of transcendentalism, contemporary to the time the book was written, can be seen anyplace where nature is to be found, notably in the forest scene, between chapters 16-19. Nature is beautiful and mysterious, and far more "good" than the religions offered in the book.
Also, through my reading of the book, I noticed that witchcraft was a big part(referring to Mistress Hibbins). Not only was this around the same time as the Salem Witchcraft Trials, but it was near the same area, as well. This could have played a role on the strictness of the puritan community. They might have punished Hester not only for her wrongdoing, but for the fact that many people were living in fear and also for the fact that the Puritans were affected by the accused witches, feeling a need to take the law and its punishments further into their own hands than previously shown.