I don't believe we can start with the sacrament of baptism but with understanding the covenants and how they are explained in scripture.
If you consider the Mosaic covenant as the covenant of grace than you may conclude that children are included in the covenant just as they were in the OT. To baptize infants would be a legal obligation just as circumcision was a legal obligation. Baptists tend, historically, to view the Mosaic covenant as a republication of the covenant of works, the same "do and live" covenant given to Adam in the garden.
"But like Adam they transgressed the covenant; there they dealt faithlessly with me." Hos 6:7
The covenant promised in Gen. 3:15 was confirmed to Abraham in Gen. 17. Christ makes this promise a reality, "for this is my blood of the covenant, which is poured out for many for the forgiveness of sins." Matt. 26:28 This 'new covenant' was not enacted until Christ's death, covenants being confirmed by blood and the "death of the testator."
"For where a will is involved, the death of the one who made it must be established." Heb. 9:16
So I guess you have to decide if the Mosaic covenant is a covenant of grace or a covenant of law and works. I find Presbyterians work from baptism backward and have created a covenant theology to support their position on baptism. Beeke notes in his Puritan Theology that many of the Divines in attendance at the Westminster Assembly held to a view of republication, their view being similar to that of the Baptists, it was not considered orthodox.
A. W. Pink Commenting on
Hebrews 8.6:
This more excellent ministry
Christ is here said to have obtained. The way whereby the Lord Jesus entered on the whole office and work of His mediation has been expressed in
Hebrews 1:4 as by inheritance: that is, by free grant and perpetual donation, made unto Him as the Son
The ministry of the old covenant was powerless, it never obtained anything and only looked forward to the promised Messiah.
For if that first covenant had been faultless, then should no place have been sought for the second.
Hebrews 8.7
The covenant which is here referred to is that into which Jehovah entered with Israel at Sinai: see
Exodus 19:5; 34:27, 28;
Deuteronomy 4:13. Israels response is recorded in
Exodus 19:8, 24:3. It was ratified by blood:
Exodus 24:4-8. This was not the first covenant absolutely, but
the first made with Israel nationally. Previously,
God had made a covenant with Adam (Hos. 6:7), and in some respects the Covenant at Sinai adumbrated [adumbrated: To give a sketchy outline of; To prefigure indistinctly; foreshadow.]
it, for it was chiefly one of works.So too He had made a covenant with Abraham, which in some respects adumbrated the Everlasting Covenant, inasmuch as it was one purely of grace. Prior to Sinai, God dealt with Israel on the basis of the Abrahamic covenant, as is clear from
Exodus 2:24; 6:3, 4. But it was on the ground of the Sinaitic covenant that Israel entered Canaan: see
Joshua 7:11, 15;
Judges 2:19-21;
1 Kings 11:11;
Jeremiah 34:18, 19.
Pink asks the question,
Wherein lay its faultiness?
It was wholly external, accompanied by no internal efficacy. It set before Israel an objective standard but supplied no power to measure up to it. It treated with men in the flesh, and therefore the law was impotent through the weakness of the flesh (Rom. 8:3). It provided a sacrifice for sin, but the value thereof was only ceremonial and transient, failing to actually put away sin. It was unable to secure actual redemption. Hence because of its inadequacy, a new and better covenant was needed.
Benjamin Keach was an early Baptist who wrote about the covenants:
True, there was another Edition or Administration of it given to Israel, which tho it was a Covenant of Works, i.e. Do this and live, yet it was not given by the Lord to the same End and Design, as the Covenant was given to our First Parents, viz. It was not given to justifie them, or to give them eternal Life; For if Righteousness had been by the Law, then Christ is dead in vain, Gal. 2.21. And again saith Paul, For if there had been a Law given, that could have given Life, verily Righteousness should have been by the Law, Gal 3.21. But indeed, it was impossible Life, Justification, or Righteousness, could be by the Law, or by any Law, because Man hath sinned, and is now unable to answer the Law of perfect Obedience, all have sinned and come short of the Glory of God: We must therefore now be justified by the Grace of God, through the Redemption which is in Jesus Christ; but tho Man had lost his Power to obey, yet God hath not lost his Poer to command. Therefore, as Dr. Owen shews, it (was added or) revealed in the Wisdom of God, as instructive; as also, to shew the Excellency of that State and Condition, in which we were created; with the Honour that God put upon our Nature: from whence Directions unto a due Apprehension of God and ourselves, may be taken or derived. It served to shew what a Righteousness Man once had, and by his Transgression lost and also what a Righteousness tis, which the Holiness of Go doth require, in order to our Justification in his sight.
Benjamin Keach on the covenant of works with Israel | Particular Voices
I hope that helps. A lot has been written about the covenants from a Baptist perspective in the last 5 years. If you want a list of titles I can make another post.
Yours in the Lord,
jm