If a proselyte was converted on the eve of Passover, - Beth Shammai maintain: he performs tebillah [ritual washing] and eats his passover-offering in the evening; while Beth Hillel rule: one who separates himself from uncircumcision is like one who separated himself from a grave. (Pesachim 92a)At the end of the passage above, the Soncino Talmud has a commentator's footnote on the purifications of one who separated himself from a grave (contact with death):
He must be be sprinkled with the water of purification on the third and seventh days after the circumcision; hence he is not yet fit in the evening. (editorial footnote to Pesachim 92a)
The first comment's inclusion in the Mishnah rather than the Gemara, along with its reference to the schools of Hillel and Shammai, suggests an early historical date for Jewish baptism of converts. The Christian practice is then the inheritor of the Jewish practice.
The Talmud also speaks of the Jewish practice of baptizing very young children, and special status is given to those who grew up in the faith, having been baptized at under 3 years of age. It stipulates
A minor proselyte is immersed by the direction of the court. (Kethuboth 11a)The Talmud also here discusses opinions if the baptized child rejects conversion upon coming of age.
The general approach of Judaism regards conversion as a family affair, with the children being considered proselytes along with the parents and being baptized along with them. There are some differences to the Christian pedobaptist practice in that baptism was not the normal mode of entry to the covenant, but only the mode of entry for those not born into the covenant. Children born to parents who were already baptized were considered part of the covenant already.
On the matter of acting on behalf of children and securing their baptism, it was reasoned,
That it is an advantage to him and one may act for a person in his absence to his advantage? Surely we have learned this already: One may act for a person in his absence to his advantage, but one cannot act for a person in his absence to his disadvantage! (Kethuboth 11a)
In light of the pre-existing Jewish practice of infant baptism along with a proselyte household, the household baptisms recorded in the New Testament have a historical context by which they should be understood and interpreted. Household baptism, which included infant baptism, was an established practice and was continued in the New Testament. In light of the historical context of household baptism, infant baptism cannot justly be said to have no Scriptural warrant in the New Testament.
I am deeply indebted to Uuras Saarnivaara's Scriptural Baptism, in which he begins to develop this argument without Talmud citations, but gives enough of the text of the quotes for me to chase them down in an on-line searchable Talmud.