Scout is influenced by her brother in a variety of ways.
First, Scout wants to do what her brother does, even at the risk of her own safety. For example, Scout goes to the Radleys late at night with Jem and Dill even though she doesn't want to. She does it because the boys start to call her a girl and treat her like she is scared. She refuses to be pushed around or out of their circle of friendship. So, in spite of being afraid, she goes.
Secondly, because Scout doesn't have a mother, and spends most of her time with Jem, she likely gets her "tomboy" tendencies from this relationship. The two like to play in their treehouse together and the two are excited about their air rifles from Uncle Jack at Christmas together. These boy activities influence Scout to act more like a boy than the girl that she is forced to play by the end of the story at the Missionary Society Circle.
Scout is affected by Jem's every emotion. When she can tell he is being dared by Dill to go touch the Radley's house in the beginning, Scout senses his apprehension but knows he won't back down from a dare. Throughout Jem's quiet times and moody times Scout knows to leave him alone because he is either thinking about his mother or the trial.