How is this a story about responsibility and duty? What happens when someone fails in taking responsibility for his/her actions?

Certainly Victor had a responsibility to his creation just as a parent has a responsibility to their child. Victor's duty to the creature didn't end with its creation. Victor abdicated that responsibility and unleashed hell in his world. Similarly, when people do not live up to their parental responsibilities, the child can have many troubles in society. This isn't to say they would go on a cross-country killing spree but they might be prone to certain negative emotions.