What's the main main-stream Protestant view on how much it is a duty for Christian children to believe in what their Christian parents believed? What Scriptural basis do Protestants put forth to support their views?
I do not believe this is only about Protestantism but it is about Christianity in general and even other religions also.
Every normal person in this world, regardless of which religion we follow, we all have inherent wisdom to lead our children to the right path which we believe to be true. A Muslim would train his child in the paths of Islam because he believes that Islam is the true religion. A Hindu would also do the same just as what a normal Christian would do. Any religious parent will teach their children about their religion.
In any case, the Bible still have something to say about children. Jesus welcomed the children and even encouraged others to bring them to Him.
These words of Jesus and other passages from the Bible became an inspiration to many Christian denominations to practice Infant Baptism. It is believed that by baptizing the children, even if they die at a young age, they might enter Heaven. Many churches also encourage parents to bring their children to the church and special Sunday School programs are designed for children.
It is very important to train the children in the way of the Lord when they are young. This will be a guide throughout their whole life.