May we baptize children?
Yes. We believe children are under the atonement of Christ and as heirs of the Kingdom of God are acceptable subjects for Christian Baptism.
Catechism for the Global Methodist Church, question 41
15They were bringing infants to him so that he might touch them, but when the disciples saw this they began to rebuke them. 16Jesus summoned them and said, "Permit the children to come to me and do not hinder them, for of such is the kingdom of God. 17Truly I say to you, whoever does not receive the kingdom of God like a child will never enter it."
44While Peter was still speaking these words, the Holy Spirit fell upon all who were hearing the message. 45The faithful from the circumcision were amazed, as many as came together with Peter, that also upon the Gentiles the gift of the Holy Spirit has been poured out. 46For they were hearing them speaking tongues and magnifying God. Then Peter answered. 47Surely no one can withhold the water for these people not to be baptized, who have received the Holy Spirit just as we have. 48He ordered them in the name of Jesus Christ to be baptized. Then they asked him to remain some days.
15When she was baptized, along with her household, she urged us, saying, "If you have judged me to be faithful to the Lord, come into my house and stay." And she prevailed upon us. 30Then he brought them out and said, "Sirs, what must I do to be saved?" 31They said, "Believe in the Lord Jesus, and you will be saved, you and your household." 32They spoke the word of the Lord to him and to all who were in his house. 33He took them at that hour of the night and washed their wounds; then he and all his household were baptized at once. 34He brought them up into his house, set food before them, and rejoiced with his whole household because he had believed in God.
8Crispus, the ruler of the synagogue, believed in the Lord together with all his household, and many of the Corinthians who heard believed and were baptized.
16I did baptize also the household of Stephanas; beyond that, I do not know whether I baptized anyone else.

Children may be baptized. Article VI of the Confession of Faith teaches it: "We believe children are under the atonement of Christ and as heirs of the Kingdom of God are acceptable subjects for Christian Baptism." Article XVII of the Articles of Religion says the same thing in fewer words: "The Baptism of young children is to be retained in the Church." Christians have argued over this practice since the Reformation, and the Methodist church has never wavered on it.
Children are under the atonement of Christ because Christ died for all. Article VIII of the Confession of Faith calls his offering on the cross "the perfect and sufficient sacrifice for the sins of the whole world," and ΒΆ102 of the Book of Doctrines and Discipline describes God's preventing grace as reaching people "even before we are aware of our need for God." A child is covered by that sacrifice before the child can ask for it. Children are also heirs of the Kingdom of God, and Jesus said so himself. When infants were brought to him and the disciples objected, he said, "Permit the children to come to me and do not hinder them, for of such is the kingdom of God" (Luke 18:16). The kingdom belongs to them, so the church does not withhold the sign of the kingdom from them.
The apostolic church baptized whole households. Lydia "was baptized, along with her household" (Acts 16:15). The Philippian jailer heard that salvation was for "you and your household," and his whole household was baptized that same night (Acts 16:31, 33). Crispus "believed in the Lord together with all his household" (Acts 18:8), and Paul wrote that he "did baptize also the household of Stephanas" (1 Corinthians 1:16). At Cornelius's house Peter asked, "Surely no one can withhold the water for these people" (Acts 10:47). God has claimed the children too, so the water is not withheld from them.
Methodist families can bring their children to the font with confidence. Baptism is an act of God's grace toward a child who cannot yet respond, and God's grace does not wait for a response. Article VI also says that such children "become the special responsibility of the Church," a charge the congregation accepts aloud at every infant baptism. What that responsibility requires is the subject of the next question.