Responsibility is a behavior that is learned. Part of parenting is to teach kids to be responsible. Giving them reponsibilities and holding them accountable will lead to responsible adults.
First of all we need to define responsibility. Responsibility is something we are held accountable for. Throughout the next few steps I want you to change your vocabulary from chore to responsibility, from homework to responsibility, and so on. This helps the child learn the term and associate there tasks to just that "responsibility".
Secondly you will need to find something your child will be held accountable for. This can begin at an early age. Give tem a daily chore around the house. Be sure to assign your child something age appropriate. They may need to be reminded at first but as they begin to complete their task on their own, let them know you are proud of their responsible behavior.
The older the child becomes the more responsibilities you can give them. Consider they will have responsibilities of their own, like homework, having the gym shoes ready on gym days. Then comes the car and being home by curfew.
Reward your children when you see that they are taking care of their responsibilities. Take them to the park, give them a sticker, an extra half hour before curfew, anything you would normally due, but now it's because they are being responsible. Hold them accountable for not taking care of their responsibilities. You can take away some of their play time until they have taken care of their responsibilities. Let them suffer the natural consequences that come from not being responsible. Don't run their homework or gym shoes to school if they forgot. It is their responsibility.
Lastly model good responsible behavior. Children learn from their best teachers and that is their parents.