Responsibility is the trait of owning what is ours: our duties, our choices and our effect on others. It differs from blame. A responsible person asks, what part is mine, and then acts on it. Such people are easier to rely on, and they often feel more capable themselves, since they are not waiting for others to change before they move.
Gita 4.13 says social duties arise from qualities and actions, so focus on developing character and fulfilling responsibilities rather than identifying with social labels. Gita 2.31 asks us to remember our own duty and not waver when it is righteous, and Gita 3.8 says performing rightful duty is better than idleness. Gita 3.5 says action is inescapable, so the real question is how to act wisely, and Gita 18.9 describes duty done simply because it ought to be done, without attachment to results, as renunciation in goodness.
The Gita also widens the circle. Gita 3.20 says even the wise continue to act to hold society together and guide it, so let your work serve a broader good. Gita 3.22 shows even the Lord, who has nothing to gain, engaging in action, which models acting for its own sake. Gita 14.15 shows how inner tendencies shape future experience, so treat your habits of mind as a long-term investment and gently redirect restless or careless tendencies before they harden.