Does the monster have any responsibility to his society in the book? Why?
The monster does not have a society. He pleads with Victor to create a society for him...a female companion...but Victor changes his mind. Had Victor done this, the responsibility the creature agreed to was to completely live outside of human society never again to be seen or cause fear or harm. However, with the destruction of the promised female creature, the monster vows to "be there on your (Victor's) wedding night" and to make Victor's life as miserable as his own. In so far as that responsibility goes, he is profusely successful.
As far as social responsibility as we define it for human beings, the monster/creature has none. He is not human. He is constructed from human parts, but his birth and creation is the first of its kind with the exception of Adam. There was none of the typical conception, gestation period, and birth as humankind has experienced.
To his credit, the creature has kept his promise. Although the depth of the pain and fear he has caused permeates lives outside of the Frankenstein family (he murders Justine and Henry), he has kept to people who are closest to Victor in order to cause him the most suffering.