But there's a difference between the banning in Brave New World and what we consider the term to mean today (or what it meant in 1984). Most World State citizens couldn't read at all so banning books was done so they wouldn't be tempted to learn. The government wasn't taking something away as much as they were simply not giving the population access to it.

Books were essentially like illegal drugs in our society. The Government had decided they were harmful to the citizenry and attempted to restrict them from coming in. But it's clearly a fairly minor thing (again equivalent to our drug policy where most personal use offenders get drug treatment rather than punishment).

For example, John the Savage is allowed both to quote Shakespeare and to interact with those who come to the reservation. Yet there's no Government repression and it's certainly not like in 1984 where you get a lobotomy. So the banning of books really reinforces the cartoon's point because what the government has done is to convince the population that books are bad for them and then offered the ban as a way to keep the population from being harmed by them.