Being of a confessional church, the Westminster Larger Catechism provides the following summary of the biblical doctrine for the baptism of infants.
Q. 166. Unto whom is baptism to be administered?
A. Baptism is not to be administered to any that are out of the visible church, and so strangers from the covenant of promise, till they profess their faith in Christ, and obedience to him, but infants descending from parents, either both, or but one of them, professing faith in Christ, and obedience to him, are in that respect within the covenant, and to be baptized.
The Westminster Confession of Faith in chapter 28 section four adds:
“Not only those that do actually profess faith in and obedience unto Christ, but also the infants of one, or both, believing parents, are to be baptized.”
Proof texts sited: Gen. 17:7–14; Gal. 3:9–14; Col. 2:11–12; Acts 2:38–39; Rom. 4:11–12; Matt. 19:13; Mark 10:13–16; Luke 18:15–17; Matt. 28:19; 1 Cor. 7:14.
This view stresses covenant continuity as opposed to some unknown and unspoken reason for halting the practice. The burden of proof resides more on those who would say we are, in some manner, commanded to not baptize children.
“Cornelius’s household…is there evidence that his household included infants or real young children, or is that an assumption that is being made?”
It is an educated assumption based on history, his status and because all his slaves would reside as part of the household, along with their children. It is, as the question says, assumed that there would normally be some small children and/or infants about this household.
Given the inclusive language of the text, it is a far greater assumption to propose that the household baptism did not have any small children or infants present. It is also problematic to the text to insert the assumption that the baptism was limited to only professing believers or that some age of accountability limited various children from participating. R. Scott Clark in a paper on the topic states:
“To exclude the children of believing parents from the sign of admission to the visible covenant people or to say that God no longer wishes children to be considered a part of the visible community of God’s people is no mere change in circumstance but rather a radical change in God’s way of dealing with his people.”
The exclusion of children would be a questionable change (at least) to a people familiar with God’s covenant workings. R. Scott Clark restates the question about continuance in the same paper:
“Q. How Can We Baptize Children Who Don’t Understand What is Happening to Them?
A. Did the babies circumcised under Abraham and Moses understand what was happening to them? Of course not. How were Abraham, Isaac and Jacob saved? By grace alone, through faith alone, in Christ alone. The fact God require children of the believers to understand the sign of admission to the visible covenant community before it was given, doesn’t mean that they did not need to understand it as they grew up. They certainly did. The same responsibility rests with every Christian today. Every time Christians come to the Lord’s table, they renew the covenant, receive the promise of the Gospel again, take up their oath of obedience to God and renew their baptism.
In fact, every complaint raised against Covenant baptism can be raised against covenant circumcision. If those complaints were invalid for circumcision, they are invalid for baptism.”
See also these other questions raised against infant baptisms as asked and answered by Clark at: http://clark.wscal.edu/baptism.php
Isn’t Repentance and Faith Required before Baptism?
Isn’t Faith Necessary for Entering the Christian Life?