This question could be answered through the views of different male and female characters as far as what they claim their responsibilities to be.
In the novel there is a lot of weight given to chivalry, gallantry, and the ideal of a marriage with a solid foundation provided by males. Males are expected to be providers, protectors, and loving supporters of their wives. At least, that is what the women would hope for.
However, we see in the character of Sir Walter that he has very different views of this ideal. First, he squandered his fortune much like Mrs. Smith's late husband did. He could have cared less about the security of his family. Instead, he persisted in the ideal of the eternal wealth of the aristocracy and lived up to its standards rather than in a sensible and protective way towards his daughters.
What individuals owe to their families is devotion, time, and love. In the story, we see how Lady Russell immediately took the nurturing role of the deceased Lady Elliot to protect Anne and Elizabeth. She acted as a mother would.
Sir Walter, contrarily, did not act like a father should. He lived for his own hedonism and for simple looks. However, this didn't make him a bad person. It simply showed that his generation was beginning to dwindle in the midst of a changing, developing and industrialized society in which the middle class was beginning to become wealthy, and the old ranks whose finances were decreasing had to come to terms with "mixing up with the bourgeoisie".