Responsibility is a combination of a willingness to face necessary work, and the ability to accept the burden of one's own care. It is becoming increasingly difficult to teach children to be responsible when the world around them is so willing to excuse them if they fail to develop that quality.
Much of it starts with work. Teaching them to do self-care when young, teaching them to do more than that as they get older. But teaching them how does not always guarantee that they will DO!
The other side to responsibility is to allow them a consequence if they do not. Letting them experience the result of not having lived up to their responsibility.
Children learn to be responsible by being given age appropriate, and gradually increasing tasks and expectations from the time they are able to first accept them. In order for them to learn to do the tasks necessary, they must be taught, step by step, as they grow.
When thorough training is combined with gradually increasing expectations, you give your child the best possible chance of growing into a responsible adult. While there are no guarantees, and some children take longer to learn than others, the goal of giving them the best chance is a good one, and an important one for parents to set.