My working definition of responsibility: the ability and desire to take ownership of your actions and their consequences, including the ability and desire to refrain, temporarily or indefinitely, from doing something that will bring about negative consequences to others, including yourself
Responsibility in non-monogamy means taking the necessary precautions to prevent unwanted pregnancies and the spread of disease, and willingness to deal with the aftermath of any such occurring. It also means openly discussing these things with all partners and refraining from sexual activities until such agreements are made that everyone can be comfortable with, and refraining from using alcohol and drugs that can impede judgment, especially with new partners.
The physical stuff is easy compared with the emotional stuff. To me, responsibility in non-monogamy is the opposite of "I will do what I want to do, and you need to deal with it or get going", or "I can't take responsibility for your feelings". It involves taking ownership of one's needs, not disguising them as lacks of another, as in "If only you would..." or "Since you are not willing to give this to me, I need to go outside of our relationship to seek it". It also involves the courage to end or alter relationships that bring more pain than joy, even if it means being alone for a while.
The ethical aspect of non-monogamy for me involves honesty and bravery in feeling and communication; loyalty to existing partners and commitments; respecting others as adults with feelings, thoughts and powers of their own; keeping onto one's word and agreements; the desire to bring joy to others, not only to oneself; and the ability and desire to hold onto these values even when they are inconvenient or hard to follow through.
Me: bi female in my twenties