Tip-toeing through this topic can be difficult because the arguments for and against it can become quite theologically deep. There are different opinions held by different Christian denominations - each claiming to have the clear teaching of scripture. But there are basically only two responses to whether or not it is alright to baptize infants: yes and no. Let's briefly look at both.
Baptizing infants has been practiced for a very long time in Christianity. Within the scope of this practice are denominations that insist that the baptism itself brings the infant into a salvation relationship with God. On this, CARM strongly disagrees. Baptism does not save anyone. Baptism is a sign, a covenant sign, of identification with the Christian gospel. On the other hand, there are denominations (usually Presbyterian) that baptize infants not for the purpose of saving them but for covenantal relationship. The logic used by these groups is basically as follows: 1) infants were circumcised and entered into covenant relationship with God in the Old Testament (Gen.12-17; 2); circumcision did not save the child; 3) the Abrahamic covenant under which the infants were circumcised is still valid since the Abrahamic Covenant is called the gospel in Gal. 3:8; 4) the Abrahamic covenant, which is still valid and which is the gospel, included infants; and, therefore, infants should still be included in the same covenant; 5) we see evidence of whole households being baptized in Acts (Acts 16:15). Therefore, at least some of these households had to include infants, and a good Jewish convert to Christianity would not exclude his infant children from covenantal relationship with God.
In contradiction to this, those who are opposed to infant baptism usually mention the fact that there are no explicit instances of infants being baptized in the New Testament. If anything, we are told to believe and be baptized (Acts 8:13; 18:8). They ask how can an infant believe, so he/she can then be baptized? These questions are a worthy concern.
Both sides have good arguments for their position, answering the other's objections, and presenting their case. Nevertheless, one's presuppositions will govern how he/she sees the argument. But, one thing needs mentioning: since there are good arguments on both sides and since there are good and godly Christians on both sides of the argument, we need to be gracious to those who differ in this issue. Notice what it says in Rom. 14:1-5,
"Now accept the one who is weak in faith, but not for the purpose of passing judgment on his opinions. 2 One man has faith that he may eat all things, but he who is weak eats vegetables only. 3 Let not him who eats regard with contempt him who does not eat, and let not him who does not eat judge him who eats, for God has accepted him. 4 Who are you to judge the servant of another? To his own master he stands or falls; and stand he will, for the Lord is able to make him stand. 5 One man regards one day above another, another regards every day alike. Let each man be fully convinced in his own mind."
Notice how Paul allows us to have differences of opinions on various issues? His goal was to have unity in the body of Christ and at the same time to allow for differences on debatable issues.
Finally, the position of CARM is that it is okay to baptize infants if the parents of the infant are God-fearing, Bible-believing Christians who honestly see infant baptism as a covenantally faithful act--not that it saves the child. On the other hand, if the parents do not believe it is proper to baptize their child - then it is not okay for them to do it.