The New Testament includes three passages that explicitly refer to “households” being baptized:

• “She was baptized, with her household” (Acts 16:15)
• “He was baptized at once, with all his family” (Acts 16:33; 18:8 also implicitly implies it)
• “I did baptize also the household of Stephanas.” (1 Corinthians 1:16)

Many biblical passages also connect household and children (e.g., Genesis 18:19; 31:41; 36:6; 47:12; Numbers 18:11; 1 Chronicles 10:6; Matthew 19:29; 1 Timothy 3:12), and several others reference entire households being saved (Luke 19:9; John 4:53; Acts 11:14; 16:31; 18:18). 

On a Reddit AskHistorians page (“What was the average number of children a Christian family had around 0-100 A.D.?”), one of the historians stated: “Ancient Hebrew peasant families typically had 4-8 children …”

Leo G. Perdue, in his book, Families in Ancient Israel, observed:

Family households did not consist of nuclear families in the modern understanding of a married couple and their children but rather, were multigenerational (up to four generations) and included the social arrangement of several families, related by blood and marriage, who lived in two or three houses architecturally connected. …

Those who belong to the family household are mentioned a number of times in the Hebrew Bible (Genesis 7:1, 7; 36:6; 45:10; cf. Genesis 46:26; Exodus 20:8-10, 17; Deuteronomy 5:12-15, 21; Joshua 7:16-18; Judges 6:11, 27, 30; 8:20). These texts indicate that the family household was primarily a kinship system that included lineal descent and lateral extension: grandparents, adult male children and their wives and children, unmarried children, and widowed and divorced adult daughters who may have had children.

Thus, it’s not just a matter of a nuclear family (which already may have included 4-8 children), but of extended family (involving even more children), which makes it all the more likely that children would typically be present in a biblical “household.” This was before widespread contraception. Children were a blessing, both according to the biblical texts saying so, and economically, since children provided labor on the family farm or as a worker in its trade.

It’s very difficult, logically and exegetically, and in light of relevant historical knowledge, to concludes that a biblical “household” could not possibly contain small children. What we need to determine (and what I sought to do above) is the precise meaning of a biblical “household.” If they usually contained children, then the Bible virtually describes infant baptism.

The biblical case for infant baptism is an argument from plausibility or antecedent probability. The deductions made lead one to conclude that a certain state of affairs is probable, more or less, but not absolutely proven. These deductive steps with regard to infant baptism are as follows:

1. All agree that the Bible refers to entire households being baptized (because of three explicit passages).
2. It is altogether reasonable to assume that most households (especially in the ancient world) would include children.
3. The Bible specifically places children — at least eight times (see above) — within the parameters of those persons included in a household. 
4. Therefore, it is quite likely that baptisms of entire households would include baptisms of children, at least in some cases, if not in all.
5. It is quite unlikely that baptisms of entire households (granting the premise that the households can and usually do include children) would never include children.
6. Therefore, infants (in the greatest likelihood) were baptized, and infant baptism is sanctioned in Scripture and apostolic example.

Plenty of classic Protestant Bible commentaries (in their analyses of Acts 16:15) concur with the argument I have made (since, after all, the historic overwhelmingly majority position among Protestants is infant baptism):

• Barnes’ Notes on the Bible: No mention is made of their having believed, and the case is one that affords a strong presumptive proof that this was an instance of household or infant baptism. Because: (1) Her believing is particularly mentioned, (2) it is not intimated that they believed, (3) it is manifestly implied that they were baptized because she believed.
• Expositor’s Greek Testament: As in the case of Cornelius, so here, the household is received as one into the fold of Christ. ... We cannot say whether children or not were included ... but nothing against infant baptism, which rests on a much more definite foundation, can be inferred from such cases …
• Bengel’s Gnomen: Who can believe that in so many families there was not a single infant? and that the Jews, who were accustomed to circumcise their infants, and the Gentiles, to purify their infants by washings (lustrations), did not also present them for baptism?
• Pulpit Commentary: This frequent mention of whole households as received into the Church seems necessarily to imply infant baptism. The exhortations to children as members of the Church in Ephesians 6:1, 2, and Colossians 3:20, lead to the same inference.

Even critics of infant baptism deduced from the “household” passages fairly admit that it can’t be ruled out. Ellicott’s Commentary for English Readers comments on the same passage: “the utmost that can be said is that the language of the writer does not exclude infants.”

The 1909 (Protestant) Hastings’ Dictionary of the Bible (“Baptism”) arrives at the same conclusion as to the inclusion of children in baptism:

From the time of Abraham onwards the Jew had felt it a solemn religious obligation to claim for his sons from their earliest infancy the same covenant relation with God as he himself stood in. There was sufficient parallelism between baptism and circumcision (cf. Colossians 2:11) for the Jewish-Christian father to expect the baptism of his children to follow his own as a matter of course. The Apostle assumes as a fact beyond dispute that the children of believers are ‘holy’ (1 Corinthians 7:14), i.e. under the covenant with God, on the ground of their father’s faith.