God says infant baptism is wrong. Consider the following passage:
Acts 8.35-37 (KJV)
35 Then Philip opened his mouth, and began at the same scripture, and preached unto him Jesus.
36 And as they went on their way, they came unto a certain water: and the eunuch said, See, here is water; what doth hinder me to be baptized?
37 And Philip said, If thou believest with all thine heart, thou mayest. And he answered and said, I believe that Jesus Christ is the Son of God.
38 And he commanded the chariot to stand still: and they went down both into the water, both Philip and the eunuch; and he baptized him.
God has made it clear in this passage that before anyone can be baptised, they must meet this prerequisite: they must believe with all their heart that Yahshua is The Son of God. Philip knew this was true in the case of the eunuch, because the eunuch confessed it with his mouth.
In the case of an infant who cannot yet speak, how can we possibly know whether or not they believe with all of their heart that Yahshua is The Son of God? We can’t. Furthermore, in the case of an infant who is not yet old enough to understand spoken or written words, it’s clear that they cannot even understand what the words ‘If thou believest with all thine heart, thou mayest’ mean, let alone believe them.
As such, we must conclude that an infant does not meet the prerequisite which God has set for a person to be ready to be baptised in water and so if whilst we are seeking to fulfil God’s command to ’[go] … and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost…’ (Matthew 28.19), we encounter (as I have many times) someone who says “I have been baptised” and we find out after probing that this was as a baby, we know we have some explaining to do, since this baptism was completely and utterly ineffectual.
To respond to Peter Turner's question about whether the infants in Cornelius' household were rebaptised when they reached the age of majority, I would pose the following question: why do you assume there were any children in Cornelius' household that were so young that they couldn't understand the meaning of spoken (or indeed written) words, hence being too young to believe that Yahshua is The Son of God, hence being too young to be baptised?
Now let's think about the assertion above that John's baptism was invalid. If we examine the scripture quoted (Acts19.3-5), we find that nowhere did Paul say John's baptism was invalid. We actually find that it was a different baptism to being baptised in The Name of The Lord Jesus. There was nothing wrong with it. It was God's will for their lives, as He makes clear in John 7.29-30:
28 For I say to you, among those born of women there is not a greater prophet than John the Baptist;[d] but he who is least in the kingdom of God is greater than he.”
29 And when all the people heard Him, even the tax collectors justified God, having been baptized with the baptism of John. 30 But the Pharisees and lawyers rejected the will of God for themselves, not having been baptized by him.
It was God's will for them to be baptised with the baptism of John before Yahshua had died on the cross, risen from the dead and given the following instruction:
18 And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth.
19 Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:
20 Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you always, even unto the end of the world. Amen. (Matthew 28.18-20)
Nowhere in The Bible does it say anyone was baptised in the name of the Father, and of the Son, and of the Holy Ghost before Yahshua had died on the cross. I suspect this is because it took Yahshua's death on the cross to make this possible.
If the church wants to regain the favour of God that the early believers enjoyed, we need to examine how they operated and do likewise. You'll find in the book of Acts that when people believed the preaching of the apostles or an evangelist (Philip), they were baptised immediately. Salvation is urgent. People have died and gone to a lost eternity today (unless perhaps someone raises them from the dead and they get some more time in this life), to be thrown into the lake of fire, because they didn't understand the word that was preached:
18 Hear ye therefore the parable of the sower.
19 When any one heareth the word of the kingdom, and understandeth it not, then cometh the wicked one, and catcheth away that which was sown in his heart. This is he which received seed by the way side. (Matt 13.18-19).
48 He that rejecteth me, and receiveth not my words, hath one that judgeth him: the word that I have spoken, the same shall judge him in the last day.
49 For I have not spoken of myself; but the Father which sent me, he gave me a commandment, what I should say, and what I should speak.
50 And I know that his commandment is life everlasting: whatsoever I speak therefore, even as the Father said unto me, so I speak. (John 12.48-50).
46 And why call ye me, Lord, Lord, and do not the things which I say?
47 Whosoever cometh to me, and heareth my sayings, and doeth them, I will shew you to whom he is like:
48 He is like a man which built an house, and digged deep, and laid the foundation on a rock: and when the flood arose, the stream beat vehemently upon that house, and could not shake it: for it was founded upon a rock.
49 But he that heareth, and doeth not, is like a man that without a foundation built an house upon the earth; against which the stream did beat vehemently, and immediately it fell; and the ruin of that house was great. (Luke 6.46-49).
Make no mistake: if you want to be like the man in the parable of whom The Lord said "the ruin of the house [he built] was great", when you here Yahshua's words, don't do them.
16 He that believeth and is baptized shall be saved; but he that believeth not shall be damned. (Mark 16.16).
In response to Affable Geek's comment below, to which counter case do you refer?
In response to fredsbend, thanks for the edit and formatting help.