Gen14:14 When Abram heard that his kinsman had been taken captive, he led forth his trained men, born in his house, 318 of them, and went in pursuit as far as Dan.
Abraham had a gigantic household – 300 men of fighting age, not including children or the elderly. That was a lot of cirumcisions – and problematic for those who teach that God's everlasting covenant with Abraham was merely about a genetic people group since of the hundreds circumcised, only Isaac and Ishmael were descendants of Abraham.
Not much of a notion of religious freedom in those times, so everyone in Abraham's household (probably 1000+) would have been taught the ways of Abraham's God.
In our time, certainly, it's a bit different. Even if the Brady Bunch converted to Christianity, I doubt they would have forced Alice to convert or expect her to be baptized with them. They probably would have taught her some of what they were learning...but ultimately she was more of an employee who ran her private life autonomously.
The real question, though, is what view of religious freedom was held by 1st century Jews. When Lydia converted there was no mention of her household believing, yet the bible does teach their baptism. When devout Jews like Paul and Peter – who certainly had Abraham's story memorized and woven tightly into their cultural worldview - taught the baptism of “households”, they certainly would have had a cultural understanding of “households” that was closer to Abraham's. As such, their notions of households would have included extended family – who were more likely to share a household in those days than now – and servants/slaves (including any infants of either group). If the head of the household announced that they had converted to a new religion, then the entire household would have been discipled in those new teachings.
Baptism is for disciples, just as circumcision was. While Baptists suggest that they baptize only believers, that was not true in NT times and it is not true now. The baptized NT church included the likes of Ananias and Sapphira, Simon Magus, Hymenaeus and Alexander (with their shipwrecked faith), the apostates in 1John 2:19 who left the church and thereby showed they never really believed, and guys like the one in 1Cor5 who was shacked up with his stepmom and had to be removed from the church. In the OT times, not all Israel was true Israel...and as the Bible describes the OT people of God and the NT people of God with the same language, it is no surprise that such a situation continues to be true.
I do think Jesus practices “believer’s baptism.” Jesus knows what is in our hearts. When Jesus washes us with his Spirit (when we are “baptized into Christ” in a baptism without human hands) there is a perfect 1:1 correspondence with saving faith. The same was true of circumcision - “circumcision of the heart” was a perfect correlation with true saving faith in the OT, while Jews were admonished not to trust in physical circumcision.
That is what the new covenant passages are teaching - “And they shall not teach, each one his neighbor and each one his brother, saying, ‘Know the Lord,’ for they shall all know me...” But while Baptists believe that applies to how they run their churches, they baptize non-believers (because unlike Jesus, they cannot perfectly see our hearts) and continue to recruit pastors and Sunday School teachers. I get that they mean well but you simply cannot administrate the Church like that with fallen people involved. (and don't get me started on how Baptists assume everything in the OT is the “old covenant” - Heb8:9 specifies that the old covenant is the one made with Moses. God's covenant with Abraham is fulfilled and reaffirmed in the NT. In what sense is Abraham the “father” of gentiles today? He’s not my genetic father. He’s not my spiritual father (that would be God). Romans4 and Galatians3 seem to be pointing out that the Abrahamic covenant was not just for the OT.
Eph2:11 Therefore remember that you, once Gentiles in the flesh—who are called Uncircumcision by what is called the Circumcision made in the flesh by hands— 12 that at that time you were without Christ, being aliens from the commonwealth of Israel and strangers from the covenants of promise, having no hope and without God in the world. 13 But now in Christ Jesus you who once were far off have been brought near by the blood of Christ…19 Now, therefore, you are no longer strangers and foreigners, but fellow citizens with the saints and members of the household of God, 20 having been built on the foundation of the apostles and prophets, Jesus Christ Himself being the chief cornerstone…)
While modern households include notions of DINKs and SINKs and empty-nesters, I don't believe Paul and Peter were teaching and writing with these notions in mind.
So if Baptists deny their young children autonomy in this matter (as I think they should) and drag them to church when they are young even if it's a bit unwillingly sometimes, then I think it is right to baptize their babies as they will be discipled and brought up in the nurture and admonition of the Lord (just as the people of God did in the OT.) Circumcision and baptism are both signs that point to the need for justification by faith (“that faith makes you clean”) and both are appropriate for application to infants who lack the facility to express faith.
More discussion is here if you really are hung up on the servant/slave aspect: