All Boo does is watch the neighborhood, leave trinkets for Jem and Scout, and protect them when they're attacked. Like killing a mockingbird, arresting Boo would serve no useful purpose, and harm someone who never meant anyone any harm. So over the course of the novel, killing mockingbirds are associated with the sinful, the pointless, and the cruel.
On the one hand, linking particular characters to mockingbirds reduces them to the level of animals. On the other, it says that even animals are worthy of sympathy and the respect of being left alone if they're doing the same to you. By equating killing mockingbirds with wanton destruction, the book prompts us to take a step back from knee-jerk reactions (escaped convicts must be shot! murderers must be arrested!) and ask, what benefit is there? Why do this? What does it accomplish?