For a person to act responsibly in Maycomb, Alabama in the 1930's, she must have the courage to be just in standing up for others, especially when doing so goes against the traditions and customs of the majority of the town. There is clearly an underlying, and sometimes overt, strand of racism and class politics in Maycomb in the 1930's. Atticus Finch is the character who most clearly exemplifies what it means to act responsible in To Kill a Mockingbird. He always considers the perspectives of others: who they are, what they've been through and why. Consider the moment in Chapter 23 when Atticus explains that the reason he did not retaliate when Bob Ewell spit on him was the thought that it might save Mayella one extra beating:
Jem, see if you can stand in Bob Ewell’s shoes a minute. I destroyed his last shred of credibility at that trial, if he had any to begin with. The man had to have some kind of comeback, his kind always does. So if spitting in my face and threatening me saved Mayella Ewell one extra beating, that’s something I’ll gladly take. He had to take it out on somebody and I’d rather it be me than that houseful of children out there.
Atticus also stands up for what is right, no matter what others think of him. This is why he is the perfect choice to represent Tom Robinson. Note that the title comes from a lesson from Atticus that it is a sin to kill a mockingbird, meaning that it is a sin to kill/harm anything or anyone that is innocent and does nothing but help others. Therefore it is a sin to harm or kill someone like Tom or Boo.
Speaking beyond the novel, that is to say - historically in the American south in the 1930s but in relation to Lee's novel, for a person to act responsibly, he/she must have the courage to defy racism in all its forms. He/she must have the courage to not buy in to slanderous gossip (referring to the exaggerated stories about Boo Radley). And he/she should be willing to help others. And perhaps most importantly, a responsible person of any time period in any location should consider the feelings and perspectives of others. This is a recurring theme in the novel and Scout shares an insight about this in the final pages of the book while standing on Boo's porch:
Atticus was right. One time he said you never really know a man until you stand in his shoes and walk around in them.
In considering the perspectives of others, we get to know and understand them better. Otherwise, we compartmentalize others in classes with labels and make no effort to understand who they are. The responsible thing to do is to make the effort to be unselfish, to think about the welfare of all other people, especially the mockingbirds.