Responsibility was a duty or obligation that someone had towards others or towards the tasks they are assigned.

In 2254, as a captive of the Talosians, Captain Christopher Pike expressed his frustration with the situation by throwing himself against the wall of his cell. Later, when he was experiencing an illusion of being back home in Mojave with Vina, Captain Christopher Pike recognized it as an escape from reality. It was a life free of responsibilities and frustrations. (TOS: "The Cage")

In 2369, the Prophets asked Benjamin Sisko how linear beings could take responsibility for their actions if they could not be certain beforehand what the consequence of them would be. He explained that they used past experience to help guide them. (DS9: "Emissary")

As first officer of the USS Enterprise-D, ship resources were the responsibility of Commander William T. Riker. (TNG: "Lessons")