This answer is not meant to replace the other well stated answers, but is meant to supplement/support them by addressing the following question:
Which denomination(s) believes that infant baptism is a mean of
securing the afterlife of the baby?
Catholics, Eastern and Oriental Orthodox, Anglicans, Lutherans, Presbyterians, Methodists, some Nazarenes, the United Church of Christ (UCC), and the Reformed churches all practice infant baptism.
The Catholic Church, as well as the Orthodox churches teach and believe that baptism is sacramentally regenerative. This means that baptism is not only an outward sign, but is also an inward rebirth or regeneration.
The Catechism of the Catholic Church states explicitly:
Holy Baptism is the basis of the whole Christian life, the gateway to
life in the Spirit (vitae spiritualis ianua), and the door which gives
access to the other sacraments. Through Baptism we are freed from sin
and reborn as sons of God; we become members of Christ, are
incorporated into the Church and made sharers in her mission: "Baptism
is the sacrament of regeneration through water in the word." (CCC
The Catechism continues to explain the sacrament of Baptism within the context of infant regeneration:
Born with a fallen human nature and tainted by original sin, children
also have need of the new birth in Baptism to be freed from the power
of darkness and brought into the realm of the freedom of the children
of God, to which all men are called. The sheer gratuitousness of the
grace of salvation is particularly manifest in infant Baptism. The
Church and the parents would deny a child the priceless grace of
becoming a child of God were they not to confer Baptism shortly after
birth. (CCC 1250)
The Church also emphasizes that "baptism is a required sacrament of faith."
The writings of the early Church Fathers clearly indicate that the sacrament of baptism has been thought of as necessary for the salvation of infants since the Ante-Nicene period of the Church.
"And they shall baptise the little children first. And if they can
answer for themselves, let them answer. But if they cannot, let their
parents answer or someone from their family." Hippolytus of Rome,
Apostolic Tradition, 21 (c. A.D. 215).
"Baptism is given for the remission of sins; and according to the
usage of the Church, Baptism is given even to infants. And indeed if
there were nothing in infants which required a remission of sins and
nothing in them pertinent to forgiveness, the grace of baptism would
seem superfluous." Origen, Homily on Leviticus, 8:3 (post A.D. 244).
"But in respect of the case of the infants, which you say ought not to
be baptized within the second or third day after their birth, and that
the law of ancient circumcision should be regarded, so that you think
one who is just born should not be baptized and sanctified within the
eighth day...And therefore, dearest brother, this was our opinion in
council, that by us no one ought to be hindered from baptism...we
think is to be even more observed in respect of infants and newly-born
persons…" Cyprian, To Fidus, Epistle 58(64):2, 6 (A.D. 251)
See INFANT BAPTISM
During the Reformation period of the sixteenth and seventeenth centuries, the early reformers adopted the doctrine of sola fide (by faith alone). This doctrine teaches that the only way a person can obtain salvation is by an inward assent of faith in the saving power of the blood of Jesus Christ and the imputation of his righteousness. Sola fide consequently and necessarily rejects baptismal necessity, therefore reducing the act of baptism to an outward symbolic acknowledgment of salvation.
Reformed theology teaches that since infants lack the capacity to place faith in Christ, there can be no inward assent to faith in Christ, rendering the salvific element absent from baptism. It also intended to celebrate the covenantal relationship between the infant and God.
In summary - The Catholic/Orthodox churches have taught since the apostolic era that baptism sacramentally "saves" the infant, whereas most Protestant denominations have taught since the sixteenth and seventeenth centuries that baptism is a sign and seal of God's covenant to that child.