9 And Jesus said unto him, This day is salvation come to this house, forasmuch as he also is a son of Abraham.
10 For the Son of man is come to seek and to save that which was lost.
31 And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
Obviously, if a person believes, there is no guarantee that their family will be saved. However, in a sense salvation has come to their house. Their family are brought into a new relation of blessing and privilege. These blessings extend not only to increased opportunity to hear the Gospel, but also transformed family relationships. For God created marriage and family and when family is brought into God's transforming and renewing work, then family members can experience an increased knowledge of the goodness of that created insitution of family.
The Gospel is good news for family:
3 The aged women likewise, that they be in behavior as becometh holiness, not false accusers, not given to much wine, teachers of good things;
4 that they may teach the young women to be sober, to love their husbands, to love their children,
5 to be discreet, chaste, keepers at home, good, obedient to their own husbands, that the word of God be not blasphemed.
1 Timothy 3
12 Let the deacons be the husbands of one wife, ruling their children and their own houses well.
22 ¶ Wives, submit yourselves unto your own husbands, as unto the Lord.
23 For the husband is the head of the wife, even as Christ is the head of the church: and he is the saviour of the body.
24 Therefore as the church is subject unto Christ, so let the wives be to their own husbands in every thing.
25 Husbands, love your wives, even as Christ also loved the church, and gave himself for it;
26 that he might sanctify and cleanse it with the washing of water by the word,
1 Children, obey your parents in the Lord: for this is right.
2 Honor thy father and mother; which is the first commandment with promise;
3 that it may be well with thee, and thou mayest live long on the earth.
4 And, ye fathers, provoke not your children to wrath: Col. 3.21 but bring them up in the nurture and admonition of the Lord.
The Gospel makes the family a sphere of blessing, even if not all within a Christian family are born-again believers themselves. Thus, even though we do not know for certain whether any of the household baptisms of Acts included infants, the concept of a household salvation is very meaningful and entails blessing even for those who do not as yet believe.
In a real sense, the families of believers are said to be sanctified:
1 Corinthians 7
14 For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else were your children unclean; but now are they holy.
Notice that only the children are said to be holy. Not the unbelieving spouse. The children of a Christian parent are in a special relationship to God. This raises the question of why these children should not be baptized. If they are holy and clean, why leave them unclean in a symbolc sense by leaving them unbaptized? Why leave them sybolically outiside the Ark (1 Peter 3:20-21) or in Egypt, the place of this wolrd by refusing to let them cross the Red Sea?
It is important to recognise that the Church in the New Testament is presented as a spiritual family:
1 John 2
9 He that saith he is in the light, and hateth his brother, is in darkness even until now.
10 He that loveth his brother abideth in the light, and there is none occasion of stumbling in him.
11 But he that hateth his brother is in darkness, and walketh in darkness, and knoweth not whither he goeth, because that darkness hath blinded his eyes.
12 ¶ I write unto you, little children, because your sins are forgiven you for his name's sake.
13 I write unto you, fathers, because ye have known him that is from the beginning. I write unto you, young men, because ye have overcome the wicked one. I write unto you, little children, because ye have known the Father.
14 I have written unto you, fathers, because ye have known him that is from the beginning. I have written unto you, young men, because ye are strong, and the word of God abideth in you, and ye have overcome the wicked one.
1 Timothy 5
1 Rebuke not an elder, but entreat him as a father; and the younger men as brethren;
2 the elder women as mothers; the younger as sisters, with all purity.
If we understand the Church as a community built along family lines, this should lead to the inclusion of infants within this earthly sphere of blessing and standing, even if those infants are not yet brought within the spiritual reality of that one Body of Christ. I think the notion of a redeemed family community is also reinforced by the reference to a holy nation in 1 Peter 2:
1 Peter 2
9 ¶ But ye are a chosen generation, a royal priesthood, a holy nation, a peculiar people; that ye should show forth the praises of him who hath called you out of darkness into his marvelous light:
I do not want to go into the analogy with circumcision here, but I think the idea of a holy nation gives a nod in that direction.