Does infant baptism determine the child's afterlife, according to practitioners?

Below is a quote from Martin Luthers sermon on infant baptism. He did teach that infants do receive faith in Christ during the Baptism , as well as when Jesus blessed them

37. Tell me also, what kind of reason had the little children whom Christ embraced and blessed, and upon whom he bestowed the kingdom of heaven? Were they not still without reason? Why does he command to bring them to him and then bless them? Where did they get the faith which makes them children of the kingdom of heaven? 46. In short, the baptism and consolation of children lie in the word: “Suffer the little children to come unto me; forbid them not; for to such belongeth the kingdom of God.” He has spoken this and he does not lie. Therefore it must be right and Christian to bring little children to him. This can only be done in baptism. So also it must be certain that he blesses them, and bestows the kingdom of heaven upon all who come to him, according to the words: “To such belongeth the kingdom of God.”

I'll address the two main questions from a Roman Catholic perspective:

Question 1: Does baptizing the baby have anything to do with the destination of the infant’s soul?

Short answer: Yes.

Long answer:

1. Baptism is normally a prerequisite for entering heaven, BUT
2. There are other forms of baptism besides water baptism, AND
3. At least one of those (the baptism of blood) is available to infants -- in fact the Church has a feast day for the infant martyrs of Bethlehem killed by Herod (Matthew 2:16); THEREFORE
4. Water baptism is not absolutely necessary for heaven, and MOREOVER
5. It's within the bounds of Catholic orthodoxy to hope for the beatitude of infants who die without water baptism (cf. CCC 1261)

All that being said, #1 is the most important point on a practical level. Catholic parents are encouraged to have their babies baptized as soon after birth as reasonably possible. For a healthy baby this means a baptism with full ceremony in a church, but Pope Pius XII exhorted midwives to be ready to baptize infants immediately in the context of a birth-related medical emergency, because:

The Church does not know of any means other than Baptism that assures entry into eternal beatitude (CCC 1257, emphasis mine).

Question 2: If the child dies before knowing good and evil, does the baptized infant go to heaven?

Yes! In fact in the traditional Roman liturgy the vestment color for the funeral of a baptized infant is white, which is normally worn on the feast day of a saint. And it's not uncommon for Catholic parents whose young baptized children have died to venerate those children as they would venerate any other saint.

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 1213)

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.