I am a Cumberland Presbyterian, a small denomination that broke off from Presbyterian US in 1810 based on several tenants, including the Calvinistic beliefs (I do not know what their teaching is now) of predestination and that children dying in infancy are not saved. Our “Confession of Faith” section 5.19 reads as follows:
5.19 The sacrament of baptism is administered to infants, one or both of whose parents or guardians affirm faith in Jesus Christ and assume the responsibilities of the covenant, and to all persons who affirm personal faith in Jesus Christ who have not received the sacrament.
Scripture references talk about baptizing, not only the individual, but also their household. These references are Acts 16: 14-15,32-33; I Corinthians 1:16.
Also, I heard a Presbyterian PCA minister talk about infant baptism as an issue of “it depends on where you want to start”. Remember that babies were taken to the temple to be dedicated in the Old Testament, Samuel being the best example I can cite off the top of my head.
Children baptized in the CP church are baptized into the covenant community, but still must make their own decision and profession of faith. We believe in the doctrine that “whosoever will” repent of their sins and accept Christ as their savior will be saved.
Our church believes in believer’s baptism and the dedication of infants and toddlers. The church then is held responsible, along with parents, to see that the children are brought up in the nurture and admonition of the Lord. (Leading by example, I guess.)