Infant baptism is proved: from the command of Christ.
III. (1) From the command of Christ. This was commanded by Christ, if not in so many words (kata to rhēton), yet equivalently and as to the sense: “Go ye, and teach all nations, baptizing them” (Mt. 28:19, 20). It is evident (a) because he who commands all nations to be baptized also commands infants to be baptized; for a precept concerning a genus includes all its species. (b) The design of Christ is to teach the manner of collecting and conserving the church in the world until the end of time and to prescribe it to the apostles and their successors. Now as the church consists of infants as well as of adults, so that manner has reference to both, but according to the condition of each: that adults should be taught before they are baptized, while infants should first be baptized as covenanted and Christian, but afterwards be taught in their own time. (c) From the antithesis, for “all nations” are opposed to all the Jews and them alone, as the difference between the Old and New Testaments demands, where infants as well as adults were circumcised. Therefore by parity the relation of baptism ought to be the same. (d) From the custom of the apostles, by which they declare the meaning of the precept and approve it by their practice, while they baptized entire families (i.e., parents with their children, Acts 16:15, 33; 18:8; 1 Cor. 1:16).
IV. Nor ought it to be objected that Christ puts instruction before baptism and so speaks of adults, who can be instructed, and not of infants (“teach [mathēteusate],” he says). Although Christ placed teaching before baptism, this must be referred to the baptism of adults, of which first and especially he was speaking in reference to the state of the primeval church to be constituted of adults. Therefore, here we must distinguish between the church to be constituted and constituted. In the former, adults were to be taught before they were baptized; but in the constituted, infants were to be baptized before they were taught. The predicates are such as they are permitted to be by their subjects; infants are not capable of instruction, therefore it does not pertain to them. In the meantime, the command of Christ is to be understood analogically of infants from the nature of the covenant and the perpetual use of the church from Abraham (Gen. 17:19*). The verb mathēteuein (used by Christ) properly is not so much to teach by preaching as “to make disciples.” This is done also by the administration of baptism, which is a sacrament of initiation and the first entrance into the church and family of Christ. Thus mathētas poiein (Jn. 4:1) is not simply to teach, but to make disciples and to receive into a profession of discipline; as among the Jews disciples were often made, not who were already taught, but that they might be taught. Hence the heathen to Hillelm (?Hillel): “Make me a proselyte that you may teach me.” That this is to be so understood is proved by the verb didaskein (which is added), which would seem to be tautological unless to mathēteuein is referred to something else than simple instruction and doctrine.
-Francis Turretin. Institutes of Elenctic Theology (Vol 3. The Sacraments)