First of all, you should understand the personality of your child in order to teach him or her responsibility. Try to understand the liking and disliking of your child which will help you to make him or her sensible in a hassle free manner.
Try to involve your child in household work like handing over unwashed clothes or giving a helping hand in cleaning the floor from the scattered toys. These things usually create a sense of pride in children and they start feeling older and seem a bit responsible even at the age of 3 or 4.
You should teach your children that commitment is most important thing and they should fulfil their commitment whatever the circumstances are. Ask them to fulfil their promises that they make to the parents like brushing the teeth daily, having milk regularly and the like.
Never try to support your child when he or she does something wrong. Rather you should make your child realise the wrongness of his or her action which will help your child to avoid such things in future and thus will create responsibility in his or her behaviour.
Always try to support your child whenever he or she does something good because it will motivate him or her to do that again and again which will create a sense of pride in your Child which is necessary for creating responsibility.
Do not stop your child if he or she is trying to do something which is not harmful. If he or she wants to feed the pets at home then let your child do it which will create a sense of responsibility in his or her behaviour.
You should become a role model for your child and always do such things in front of him or her that motivate your child to imitate you. When he will grow up, you will see that he is a responsible person.