Atticus influenced Scout and Jem in many ways. Let me give a few examples.
First, Atticus taught Jem and Scout what true courage was. True courage is the ability to do what is right even if you know that you are going to lose. Atticus wanted to teach Jem this lesson by his spending time with Mrs. Dubose. Atticus says:
I wanted you to see something about her—I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It’s when you know you’re licked before you begin but you begin anyway and you see it through no matter what.
Atticus basically says the same thing to Scout. When Scout asked if Atticus was going to win the case, Atticus said to Scout that he would most likely lose the case. Scout asked why, then, he was defending him. Atticus responded by saying that it was the right thing to do to defend Tom Robinson. Courage is to do what is right no matter what the outcome.
Later on, both Jem and Scout would look back and see their father as an amazing man.
Second, Atticus also taught his children about the importance of modesty and humility. This comes out when Atticus put down the rabid dog. The children learned his nickname - "One Shot Finch." They never knew this about him, because Atticus was humble.