What method does the Whisperer use to send out messages that influence children?

The correct answer is rooted in the narrative techniques used by the Whisperer in "The Mysterious Benedict Society." The Whisperer employs the television network to send out messages that directly influence children. This utilizes the powerful reach of television, which can capture children's attention effectively and can disseminate messages broadly across a large audience.

Television as a medium holds significant sway over children’s perceptions and thoughts, making it an ideal tool for the Whisperer to manipulate and control them. It allows for auditory and visual engagement, which can be particularly impactful, especially in the context of the story where the Whisperer aims to spread ideologies in a compelling manner.

In contrast, other options like newspapers, radio, and social media are either less prominent in the storyline or do not align with the main strategy employed by the Whisperer. These other forms of communication can have their own influences, but the narrative specifically highlights the television network as the primary means used by the Whisperer to reach and influence young minds.