Atticus influences his children's lives every day through his words, but more significantly, through his actions and the example he sets for them. His primary influence is observed in how he shapes their characters. Atticus guides Scout and Jem as they adopt the values that will define them as adults.
Through his defense of Tom Robinson, as well as his direct statements, Atticus counteracts the hateful, destructive racism that surrounds Jem and Scout in Maycomb. Through his influence, the children are not "infected" with racist attitudes. Atticus works hard to make certain that Jem and Scout escape this social disease.
Atticus also influences Jem and Scout in developing in them a respect for life and the rejection of cruelty. When he teaches Jem to not kill mockingbirds, the lesson is deeper than it might appear, and it takes hold in both his children. Scout clearly understands it, as evidenced by her protective attitude toward Boo Radley in the novel's conclusion.
Finally, Atticus influences his children in their sense of basic justice and their respect for it. Tom's conviction hurts Jem deeply because he understands, and rejects, its injustice. Tom's death affects Scout deeply for the same reason. Both children have come to recognize the difference between what is just and what is cruelly unjust, and they reject injustice.