One of the main tenants of most protestant groups, including baptists (and even Catholics to some extent) is that there has to be an "age of reason" in order to receive baptism.
I can’t speak for universal Baptist belief because there really is no such thing other than a broad framework of believer’s baptism, local church autonomy and an emphasis on personal Bible reading.
However, the Baptists amongst whom I lived believed the “age of accountability” at which one could be baptized was sort of specific to the individual and was more or less determined by the ability to discern and accept/not accept Christ as saviour. Thus, you might see a precocious 4-year-old baptized and a less precocious 12-year-old baptized both at their legitimate “age of accountability.” It was generally accepted that someone who made it to his or her teen years had at some point reached his or her accountability limit.
Though an emphasis was made on praying a sinner’s prayer, honest Sunday School teachers would admit it wasn’t so much the way to be saved as it was the most obvious way, an easy marker upon which one could look in retrospect.
Baptism is understood not strictly as a memorial, but a command — in the common parlance, an “ordinance” and a necessary step in church membership. From the Baptist Faith and Message
*:Christian baptism is the immersion of a believer in water in the name of the Father, the Son, and the Holy Spirit. It is an act of obedience symbolizing the believer's faith in a crucified, buried, and risen Saviour, the believer's death to sin, the burial of the old life, and the resurrection to walk in newness of life in Christ Jesus. It is a testimony to his faith in the final resurrection of the dead. Being a church ordinance, it is prerequisite to the privileges of church membership and to the Lord's Supper.
Because it is not considered sacramental, Baptism is not thought to be a part of the process of salvation, per se, though it is considered to be a part of the normal Christian life; the walk of an unbaptized believer is in some ways a hindered one, as the unbaptized has not submitted to Christ’s demonstration nor his command to be baptized.*This wasn't the exact statement we used when I was growing up, having been edited in 2000, but I think the language about baptism was unchanged.