When studying the baptism of John, I went to the Commentary on the New Testament from the Talmud and Hebraica by John Lightfoot (1602-1675). There was a vast amount of information found under his notes on Matthew 3:6.

They baptized also young children (for the most part with their parents). They baptize a little proselyte according to the judgment of the Sanhedrim: that is, as the Gloss renders it, "If he be deprived of his father, and his mother brings him to be made a proselyte, they baptize him [because none becomes a proselyte without circumcision and baptism] according to the judgment [or right] of the Sanhedrim; that is, that three men be present at the baptism, who are now instead of a father to him."

And the Gemara a little after; If with a proselyte his sons and his daughters are made proselytes also, that which is done by their father redounds to their good. R. Joseph saith, When they grow into years, they may retract. Where the Gloss writes thus; "This is to be understood of little children, who are made proselytes together with their father."

"A heathen woman, if she is made a proselytess, when she is now big with child,--the child needs not baptism: for the baptism of his mother serves him for baptism." Otherwise, he were to be baptized.

"If an Israelite take a Gentile child, or find a Gentile infant, and baptizeth him in the name of a proselyte,--behold, he is a proselyte."

First, You see baptism inseparably joined to the circumcision of proselytes. There was, indeed, some little distance of time; for "they were not baptized till the pain of circumcision was healed, because water might be injurious to the wound." But certainly baptism ever followed. We acknowledge, indeed, that circumcision was plainly of divine institution; but by whom baptism, that was inseparable from it, was instituted, is doubtful. And yet it is worthy of observation, our Saviour rejected circumcision, and retained the appendix to it: and when all the Gentiles were now to be introduced into the true religion, he preferred this 'proselytical introductory' (pardon the expression) unto the sacrament of entrance into the gospel.

One might observe the same almost in the eucharist. The lamb in the Passover was of divine institution, and so indeed was the bread. But whence was the wine? But yet, rejecting the lamb, Christ instituted the sacrament in the bread and wine.

Secondly, Observing from these things which have been spoken, how very known and frequent the use of baptism was among the Jews, the reason appears very easy why the Sanhedrim, by their messengers, inquired not of John concerning the reason of baptism, but concerning the authority of the baptizer; not what baptism meant, but whence he had a license so to baptize, John 1:25.

Thirdly, Hence also the reason appears why the New Testament doth not prescribe, by some more accurate rule, who the persons are to be baptized. The Anabaptists object, 'It is not commanded to baptize infants,--therefore they are not to be baptized.' To whom I answer, 'It is not forbidden to baptize infants,--therefore they are to be baptized.' And the reason is plain. For when Paedobaptism in the Jewish church was so known, usual, and frequent, in the admission of proselytes, that nothing almost was more known, usual, and frequent,--

1. There was no need to strengthen it with any precept, when baptism was now passed into an evangelical sacrament. For Christ took baptism into his hands, and into evangelical use, as he found it; this only added, that he might promote it to a worthier end and a larger use. The whole nation knew well enough that little children used to be baptized: there was no need of a precept for that which had ever, by common use, prevailed. If a royal proclamation should now issue forth in these words, "Let every one resort, on the Lord's day, to the public assembly in the church"; certainly he would be mad, who, in times to come, should argue hence that prayers, sermons, singing of psalms, were not to be celebrated on the Lord's day in the public assemblies, because there is no mention of them in the proclamation. For the proclamation provided for the celebration of the Lord's day in the public assemblies in general: but there was no need to make mention of the particular kinds of the divine worship to be celebrated there, when they were always, and every where, well known and in daily use before the publishing of the proclamation, and when it was published. The case is the very same in baptism. Christ instituted it for an evangelical sacrament, whereby all should be admitted into the possession of the gospel, as heretofore it was used for admission into proselytism to the Jewish religion. The particulars belonging to it,--as, the manner of baptizing, the age, the sex to be baptized, &c.--had no need of a rule and definition; because these were, by the common use of them, sufficiently known even to mechanics and the most ignorant men.

2. On the other hand, therefore, there was need of a plain and open prohibition that infants and little children should not be baptized, if our Saviour would not have had them baptized. For, since it was most common, in all ages foregoing, that little children should be baptized, if Christ had been minded to have that custom abolished, he would have openly forbidden it. Therefore his silence, and the silence of the Scripture in this matter, confirms Paedobaptism, and continueth it unto all ages.

Fourthly, It is clear enough, by what hath been already said, in what sense that is to be taken in the New Testament which we sometimes meet with,--namely, that the master of the family was baptized with his whole family, Acts 16:15, 33, &c. Nor is it of any strength which the Anti-paedobaptists contend for, that it cannot be proved there were infants in those families; for the inquiry is not so proper, whether there were infants in those families, as it is concluded truly and deservedly,--if there were, they had all been to be baptized. Nor do I believe this people, that flocked to John's baptism, were so forgetful of the manner and custom of the nation, that they brought not their little children also with them to be baptized.

If you compare the washing of polluted persons, prescribed by the law, with the baptism of proselytes,--both that and this imply uncleanness, however something different, that implies legal uncleanness,--this, heathen,--but both polluting. But a proselyte was baptized not only into the washing-off of that Gentile pollution, nor only thereby to be transplanted into the religion of the Jews; but that by the most accurate rite of translation that could possibly be, he might so pass into an Israelite, that, being married to an Israelite woman, he might produce a free and legitimate seed, and an undefiled offspring. Hence, servants that were taken into a family were baptized,--and servants also that were to be made free: not so much because they were defiled with heathen uncleanness, as that, by that rite becoming Israelites in all respects, they might be more fit to match with Israelites, and their children be accounted as Israelites. And hence the sons of proselytes, in following generations, were circumcised indeed, but not baptized. They were circumcised, that they might take upon themselves the obligation of the law; but they needed not baptism, because they were already Israelites. From these things it is plain that there was some difference as to the end, between the Mosaical washings of unclean persons, and the baptism of proselytes; and some between the baptism of proselytes and John's baptism: not as though they concurred not in some parallel end; but because other ends were added over and above to this or that, or some ends were withdrawn.

In his Harmony of the Four Evangelists, in his notes on John 1:25, I found:

It is urged, by those that deny infants' baptism,-- that there is neither command for it, nor example of it, in Scripture, as there was for infant circumcision. Now, this consideration giveth one ready answer, if there were no other to be given:--If baptism, and baptizing of infants, had been as strange, and unseen, and unheard-of, a thing in the world, till John the Baptist came, as circumcision was, till God appointed it to Abraham,--there is no doubt but there would have been a command or example expressly given for the baptizing of infants, if God would have them to be baptized; as there was for the circumcising of infants, because God would have them to be circumcised.--But, when the baptizing of infants had been a thing as commonly known, and as commonly used, long before John came, and to his very coming, as any holy thing, that was used among the Jews; and they were as well acquainted with infants' baptism, as they were with infants' circumcision,--it doth not follow, that there needed so express and punctual a command or example, to be given for the baptizing of infants, which was well enough known already, as there needed for circumcision of infants or others; which was a thing, that, till its institution, had never been heard of, nor dreamed of, in the world...

By which we may observe, that baptism was no strange thing when John came baptizing; but the rite was known so well by every one, that nothing was better known than what baptism was: and, therefore, there needed not such punctual and exact rules, about the manner and object of it, as there had needed, if it had never been seen before. What needed it in the gospel to tell, that such or such persons were to be the objects of baptism, when it was as well known, before the gospel began, that men, and women, and children, were the objects of baptism, and were baptized,--as it is to be known, that the sun is up, when it shineth at noon-day?

The Whole Works of the Rev. John Lightfoot, D.D. Volume 4, p.407, 409