There is at the heart of the concept of responsibility the beautiful idea that it is about response (RESPONS-ability), which means it always has to do with relationships.
Responsibility is inherently mutual. Jean-Jacques Rousseau stated it well when he proclaimed that there is no meaning of responsibility that does not carry mutuality. We too often treat and confuse responsibility with obedience—as if responsibility can be imposed. There is a failure in the structure of imposition because it lacks mutuality. Although we think we give responsibility, responsibility must be TAKEN if it is to be implemented—hence its mutuality.
Responsibility has a counterpart: accountability. One reason that people resist imposed accountability is that the people at the top tell others what they are accountable for but not what they, themselves, are accountable for.
If you expect someone to be responsible and therefore accountable for OPTIMAL performance, then influence him or her to WANT to be so. An easy way to do this is to tell the person in what ways YOU will be accountable.
If you are a leader, simply explain in what ways the other person can count on you (safety, staying abreast of company policies, working environment—to name just a few). If you are a school principal, inform the staff in what ways the faculty can count on you (mutual respect, professional recognition, cooperative evaluations, etc.). If you are a teacher, inform students in what ways they can count on you (providing a classroom where students will WANT to spend their time, planning on your part to present meaningful and important lessons, engaging activities, etc). If you are a parent, the same applies (providing food, shelter, a loving relationship, someone to trust to protect their well-being, etc.)
To put the concept in easy-to-remember terms, collaboration is more effective than domination.