Households baptized
Whole households baptized (also Acts 16:15, 18:8; 1 Corinthians 1:16) beside believe-first sequences (Mark 16:16) and the promise 'to you and your children' (Acts 2:39) — the paedobaptist and credobaptist readings.
Acts 16:25-34
25But about midnight Paul and Silas were praying and singing hymns to God, and the prisoners were listening to them.
26Suddenly there was a great earthquake, so that the foundations of the prison were shaken; and immediately all the doors were opened, and everyone’s bonds were loosened.
27The jailer, being roused out of sleep and seeing the prison doors open, drew his sword and was about to kill himself, supposing that the prisoners had escaped.
28But Paul cried with a loud voice, saying, “Don’t harm yourself, for we are all here!”
29He called for lights, sprang in, fell down trembling before Paul and Silas,
30brought them out, and said, “Sirs, what must I do to be saved?”
31They said, “Believe in the Lord Jesus Christ, and you will be saved, you and your household.”
32They spoke the word of the Lord to him, and to all who were in his house.
33He took them the same hour of the night and washed their stripes, and was immediately baptized, he and all his household.
34He brought them up into his house, and set food before them, and rejoiced greatly, with all his household, having believed in God.
What each tradition says
Rabbinic / Midrashic
Jacobs and Hirsch state that a pregnant woman’s conversion included her child. Minors could be converted with their parents, or even alone, by the bet din, but they were permitted to recant when of age.
Joseph Jacobs (1854-1916) and Emil G. Hirsch (1851-1923), The Jewish Encyclopedia, vol. 10, s.v. 'Proselyte' (New York: Funk & Wagnalls, 1901-1906) — section 'Influence of the Hadrianic Persecution'
Patristic
Cyprian reports that his council rejected the view that infants should not be baptized within the second or third day after birth, or baptized and sanctified within the eighth day. They judged instead that God’s mercy and grace is not to be refused to anyone born of man.
Cyprian of Carthage (c. 200-258), Epistle LVIII (To Fidus, on the Baptism of Infants), in Ante-Nicene Fathers, vol. 5 (Buffalo: Christian Literature Publishing Co., 1886) — Epistle LVIII, §2
Catholic
The Council of Trent anathematizes anyone who says little children, having no actual faith, are not after baptism to be reckoned among the faithful, or that they should be rebaptized at years of discretion. It also condemns saying their baptism is better omitted than performed, while they do not believe by their own act, in the Church's faith alone.
The Council of Trent, Session VII (1547), The Canons and Decrees of the Sacred and Oecumenical Council of Trent (trans. J. Waterworth; London: Dolman, 1848) — Session VII, canons on Baptism, canon 13
Orthodox
The Confession of Dositheus states that even infants ought, of necessity, to be baptized. It says infants are saved, as is said in Matthew, but he that is not baptized is not saved; therefore even infants must of necessity be baptized. It adds that Acts says whole houses were baptized, and consequently the infants.
The Synod of Jerusalem (1672), under Patriarch Dositheus, The Acts and Decrees of the Synod of Jerusalem, sometimes called the Council of Bethlehem, holden under Dositheus, Patriarch of Jerusalem, in 1672 (trans. J. N. W. B. Robertson; London: Thomas Baker, 1899) — Confession of Dositheus, Decree XVI
Reformed / Protestant
Calvin argues that if the children of believers are partakers of the covenant without the help of understanding, there is no reason to deny them the sign. They should not be denied it because they cannot swear to its stipulations.
John Calvin (1509-1564), The Institutes of the Christian Religion, Book IV, ch. 16 ('Paedobaptism'), trans. Henry Beveridge (Edinburgh: Calvin Translation Society, 1845; CCEL text) — Book IV, ch. 16, §24 (on Abraham, Isaac, and the children of believers)
Evangelical
The Baptist Confession of 1688 states that the only proper subjects of this ordinance are those who actually profess repentance towards God, faith in, and obedience to our Lord Jesus. It restricts proper subjecthood to such actual professors.