When the situation with Tom Robinson came up and the case was assigned to Atticus Finch, both Atticus and his children were affected by the situation. At first, Scout did not understand it. Then, she realized the hatred with which the townspeople would regard her dad, and how she, herself, was called all kinds of names.
However, Scout also witnessed how lawyers have to stick to their clients, and defend their innocence and their rights against people, against society's opinion, and against the environment itself. Scout saw how Atticus fulfilled his duties in and outside of the courtroom, and she saw from the top of the courthouse the depth of passion that Atticus employed. She also observed and appreciated the amount of pressure he widthstood, and she even stood up for him during the incident outside of Tom's cell.
Therefore, Scout learned from her father's profession the importance of justice, and for the first time saw what Macomb was really like, hence, appreciating even with more greatness how he stood against the world as he knew it for the sake of saving an innocent man.
Having a lawyer for a father has a strong influence on Scout's character. For one thing, Atticus is not just a lawyer; he is a single parent as well. Therefore, his influence on his children is likely to be stronger than if she were being raised mostly by her mother. For another, Scout seems to have absorbed, to the extent that a kid can, Atticus' sense of justice.
Having a lawyer father also means that Scout and her brother have to rely more on themselves and each other than they would in a two-parent family. She has to make decisions on her own because Atticus isn't always there to give advice.
I think his biggest influence on her is the example of his life; he has a strong code of ethics, and Scout, whether she knows this consciously or not, is influenced by that.
In Harper Lee's To Kill a Mockingbird, Scout (and Jem) are greatly accustomed to the workings of the court system, and have a sophisticated understanding of their father's profession and the interactions between the members of the court.
Scout understands what her father does for a living, but sometimes it is hard for her to withstand the criticisms he receives for the cases he takes (like Tom Robinson's), especially when she has to hear about it from friends and relatives.
Inadvertently, Scout's father has raised his children listening to both sides of an argument before passing judgment. Without a wife, Atticus relies on his sense of fair play and the ability to place himself "in someone else's skin" in order to better understand others, which mirrors his behavior in court.
This sense of the rules of court comes into play when Scout gets into trouble with Uncle Jack over her fight with her cousin Francis. Uncle Jack immediately punishes Scout without hearing both sides of the argument. Scout draws his attention to this:
...you never stopped to gimme a chance to tell you my side of it—you just lit right into me. When Jem an' I fuss Atticus doesn't ever just listen to Jem's side of it, he hears mine too...
Atticus runs his home with many of the same rules exhibited in the courtroom, and Scout understands the equity of this, in that both sides are permitted to have a say.
It is not just that Atticus is a lawyer that affects Scout's mindset and behavior: it is seeing his role as a fair-minded, dedicated, ethical man that affects Scout. She trusts him implicitly, and Atticus is very careful to make sure that trust is not broken, more for his children's faith in him than for any other reason.
Scout knows her father is a lawyer, but what most influences her is his abiding dedication to providing his best legal counsel, regardless of the economic or racial issues. His sense of equality and dedication, which are a part of her father first, before his profession, greatly influence Scout.