The debate about baptising babies (Pedobaptism, as opposed to Credobaptism) seems to have started, according to the Didache - The Teaching of the Twelve Apostles, - somewhere in A.D.40-60.
Credobaptists argue that people should be able to believe and to declare their faith when they join the church, while Pedobaptists cite the new testament (Colossians 2:11-12) where baptism is described as a circumcision performed by Jesus, which relieves the newly baptised from further ruling of the flesh.
Since this is essential for eternal life in Heaven, Pedobaptism was considered a crucial service for the newly born, lest he or she should die before being able to prove their Good Christianity all by themselves.
If the Internet is any indication, the argument is as alive today as it has ever been, indicating that many people have nothing better to do.
I would like to add my humble addition to this debate, by noting that whenever and wherever Christianity was introduced into the world, most people who joined it were adults, and required Credobaptism to mark their transition from one group of people to the other. During the ages, as this source of new recruits was exhausted, Church authorities (urged by secular rulers,) were required to perform Pedobaptism, as a foolproof way of getting the believers to register their children and letting the rulers know how many people they ruled.