what does it mean to act responsibly in Maycomb, Alabama in 1930?

Well, considering the book shows society's racism and discrimination toward the unknown and things they don't understand, acting responsibly would be keeping out of trouble (don't wander off into unfamiliar/dangerous parts), and probably respecting elders. Respecting the elderly may be a safe bet, because children were expected to repsect older people and listen to them generally during that time and pretty much any time.