In his book Santification, Gordon H Clark takes time to discuss the issue of infant baptism:
The other subdivision of the question is a more difficult one: Whose children should be baptized? It is not at all difficult to show that a child of two believing parents should be baptized, nor even that a child of only one believing parent should be. 1 Corinthians 7:14 is sufficient. The difficulty arises when one considers the case of a child whose parents were perhaps baptized in infancy, who attend church services with some regularity, and who want their child baptized, even though they themselves have never become communicant members. Today in the United States the very large majority who are in regular attendance are communicant members. But it so happens that regular attendants who are not communicant members want their children baptized. Should the church acquiesce?
In Europe and in early America the children of baptized but non-communicant members were regularly baptized. Robert Ellis Thompson, in ‘A History of the Presbyterian Churches in the United States’ (1895, p. 14) reports: “The absence of regularly constituted sessions for the administration of church discipline, and the refusal of baptism to the children of baptized person who were not communicants, marked the local congregation as un-Presbyterian.” That is, communicant membership was not essential for the parents of infants to be baptized; and the author notes this was the rule in all the Reformed churches.
The argument was that there is a visible and an invisible Church. The members of the latter are precisely God’s elect; but many members of the former are not. Ishmael and Esau were both circumcised. Furthermore, since the promise of covenant extend to a thousand generations, the visible church today may and ought to baptize infants of unbelieving parents who want them baptized, on the basis of their ancestors’ faith. Surely not every Israelite, at any period of its disappointing history, was regenerate; yet no priest would have hesitated to circumcise the children of such parents…
At this point, Clark briefly pauses to consider the inference that the Lord’s Supper may be admitted to unregenerates, concludes Presbyterians generally follow the opinion of Edwards and Mather that it should not be so admitted, and then returns to the subject.
Now, even if it be granted that baptism may properly be administered to children of non-professing parents, the inference to a similar stance on the Lord’s Supper is fallacious…
Here he distinguishes baptism from the Lord’s Supper, offering some reasons as to why the Lord’s Supper ought not be administered to open unbelievers.
But now, beyond admission for the sake of argument, what must be said on the substantial question? Does the Bible require or prohibit baptisms to the thousandth generation? If it does, and if a generation is roughly thirty years, a thousand generation from the time of Christ would include just about everybody in the western world. Then the church should have baptized the child of an intensely Talmudic Jew whose ancestor in 50 B.C. was piously looking for the Messiah. Or, George Whitefield should have baptized Thomas Jefferson, Benjamin Franklin, and Tom Paine, as children, because one of their ancestors played a small role in the Reformation. Strange as this may seem to many, it ought to have been done if the Bible so teaches.
He now notes that “some very eminent theologians have so held,” and devotes a page to historical theology before continuing. One important statement he makes: “The view that only the children of professing parents should be baptized seems to have been the result of colonel revivalism,” and Clark is clearly sympathetic towards those ministers who had to put up with the strictness of these revivalists with respect to the recipients of baptism. He says that these standards were understandable, given that “Their pietism and evangelistic zeal led them to place great emphasis on conversion as a traumatic experience.” After a bit more historical theology, he continues:
This emotional pietism, as it demanded a particular type of experience for regeneration, tended to view the ideal church as consisting entirely of regenerate persons sharing such an experience. The logical result is the Baptist position; but in Presbyterianism it stopped short at requiring the faith of the parents who wanted their children baptized. But if it did not result in Baptists practices, it involved a change in the theology of baptism.
And thus he concludes his answer to the question posited at the beginning, proceeding to ask the meaning and accomplishments of baptism.
– Sanctification, pgs. 62-65
A similar view can be found in a letter Calvin wrote to Knox about baptizing grandchildren: http://www.baylyblog.com/2006/09/godfathers_calv.html