In the comments of the last post, Ryan asked for a positive and basic biblical argument for Baptist covenant theology. Here are the basics, and we can get more thorough as necessary. Ryan, please comment!
Though there are different versions of covenant theology, all covenant theologians emphasize the unity of three strands of biblical revelation. They say, (1) there is one moral law summarized in the decalogue; (2) there is one gospel way of salvation for all men; and (3) there is one people of God (and the one people are saved by the one gospel). There is obviously more to it than this, but this is a good place to start.
1. Prove there is one moral law. A good place to start is the book of Hebrews, specifically the new covenant revealed in Hebrews. Hebrews 8:10 records that in the new covenant, God says, "I will put my laws into their minds and write them on their hearts." The only law written immediately by the finger of God was the decalogue. This is the same law that is written on the hearts of new covenant believers. 2 Corinthians 3 shows that in the "new covenant" (1 Cor 3:6), the law of God is not written on "tablets of stone," (1 Cor 3:3), but that it is written "on tablets of human hearts" (1 Cor 3:3). The reference to "tablets" indicates the decalogue. Therefore, the one moral law of the Old Testament, summarily revealed in the decalogue, is the moral law of the new covenant. The difference between the two covenants is not a change in moral law, but the fact that in the new covenant, the law is written on the heart. This one law is what covenant theologians see as the moral precept under all the covenant administrations of the Bible.
2. Prove there is one gospel. Hebrews 9:15 says, "Therefore he is the mediator of a new covenant, so that those who are called may receive the promised eternal inheritance, since a death has occurred that redeems them from the transgressions committed under the first covenant." The death of Christ is the way men were saved under the old covenant and under the new covenant. Hebrews 9:26 says, "But as it is, he has appeared once for all [time!], at the end of the ages to put away sin by the sacrifice of himself." Hebrews 10:4 says, "It is impossible for the blood of bulls and goats to take away sins." So, men under both testaments were saved by grace through faith in Christ (cf. Gal. 3:7-9). There is no other way of salvation. This one gospel is what covenant theologians call the covenant of redemption and/or grace.
3. Prove there is one people. This is where Baptist covenant theology is distinct from both paedobaptist covenant theology and dispensationalism. The "one people of God" have always been the believing elect and have never included unbelieving infants. Dispensationalists and paedobaptists agree that the true Israel of the Old Testament to whom the promises were made included all physical/generational Israelites. But, Romans 9:6 says, "Not all Israel is Israel." That is, the whole nation of Israel is not actually Israel. Both the Old Testament and the New Testament teach that those who inherit the promises of the Old Testament are believers and that only believers in all ages are God's people. Isaiah 45:25 says that "in the Lord all the offspring of Israel shall be justified and shall glory." The book of Deuteronomy makes it clear that only those who repent of their sins and believe to the end will inherit the promises of the old covenant (Deut chs. 27, 28, 29, 30). Philippians 3:3 says, "For we are the real circumcision who worship by the Spirit of God and glory in Christ Jesus." The new covenant is made with Israel and only with Israel, "For this is the covenant that I will make with the house of Israel" (Heb 8:10). We who are in the new covenant are "Israel" along with the believing Israelites.
But, you may say, "God chose the nation of Israel." Yes, and he chose Assyria and Babylon to perform certain functions as well. National Israel was chosen as a nation, but not to inherit all of the promises God made to it. Those promises were conditioned upon the faith of individual Israelites and on the faith of the nation as a whole. Those national Israelites who failed to believe forfeited their right to the promises, and we Gentiles who believe will one day inherit all the land, health, and wealth God promised to Israel when our King comes to deliver us out of exile (we are strangers and aliens in a foreign land) and to restore us to the full blessing God promises to all who believe. We will enjoy the land of Canaan on the new earth with all its prosperity for all eternity! So, why did God choose to deal with the physical nation? Israel was chosen for a specific service and now that her service is complete, God isn't dealing with that nation as a unit any longer. God dealt with the nation of Israel as a nation to perserve the physical offspring of Abraham until Christ would come from them (Gal 3:23-27). Christ was the only reason God ever dealt with the whole physical nation. His coming was the reason for Israel's election.
Now that Christ has come, there is no need to preserve the physical nation. In fact, Israel no longer exists as a nation. The 10 tribes of Israel (north Canaan) were taken off to Assyria and have never been heard from since. Those who inhabit Canaan today are Jews, or Judeans, from the tribe of Judah who dwelt in southern Canaan. The rest of the Israelites are forever lost. But, Judah was preserved because Christ, the lion of Judah, had to come from that tribe. God allowed the rest the tribes to die off, but he preserved Judah to keep his promise to bring the messiah through that tribe. The result is that since national Israel no longer exists, the promises God made to the 12 tribes of Israel cannot be given to the group as a whole.
Furthermore, the difference between the Baptists and the paedobaptists is that unlike the old covenants, the new covenant does not include a "purely legal aspect," which can be broken. The old covenants were breakable because God did not write his law on the hearts of all covenant members. But the new covenant is different! Hebrews 8:9 says that the new covenant is "not like the covenant that I made with their fathers . . . for they did not continue in my covenant." The word "for" is critical. The new covenant is not like the old covenant for, or because, the old covenant was breakable. Everyone in the new covenant has God's laws written on his heart (Heb 8:10), knows God savingly (Heb 8:11) and has his sins forgiven (Heb 8:12). Thus, there is no room for unbelievers or reprobates in the new covenant. God unilaterally provides faith to all who are in it. Therefore, the paedobaptists are wrong to insist that infants are included. Infants were only included in the OC because there was a need to demonstrate that God was keeping his promise to bless the nations through Abraham's seed, and his seed was Christ.