Why didn't Atticus want the children to play their game or give Boo the note?
It was also important to Atticus that his Maycomb neighbors in To Kill a Mockingbird live in relative peace and privacy. Atticus could see that the children's play-acting on the sidewalk was becoming a public nuisance, and he let them know that he was not happy.
"Does this by any chance have anything to do with the Radleys?"
"No sir," said Jem, reddening.
"I hope it doesn't..."
When Atticus discovered the note on the fishing line, he put his foot down. He considered Jem and Scout's unwanted intrusion as a form of "tormenting," and until then, Boo deserved the right to "stay inside free of the attentions of inquisitive children." He told them to stay away from the house until they received an invitation.
In my opinion, this is because of how Atticus disapproves of being mean to people who are weaker.
We can see throughout the book that Atticus believes you need to treat people well no matter who they are. He does not look down on the poor whites or even on the blacks who are the lowest level of Maycomb society in the book.
If he let the kids play their game or give Boo the note, he would essentially be allowing them to pick on Boo because he is such an outsider.