There are many ways to teach responsibility... chores, allowances, learning to save your money and learning to follow through on what you say you're going to do. In my day (so very very loooong ago...) I got to write out my mother's checks for her to sign, making me feel very moneybags-cool, and very in control. With one wave of a very powerful blue-inked piece of paper, people give you things!
Now that checks are basically obsolete, I'd imagine that learning how to be responsible is more focused on learning to be a kind and helpful member of the community. Perhaps that starts with... learning to take care of something! Like, perhaps... a puppy! This family has never embraced a challenge so heartily.