There are two key passages which are used in support of this position: Acts 2:38 and John 3:5-6. Let's take them one at a time:
Now when they heard this they were cut to the heart, and said to
Peter and the rest of the apostles, "Brothers, what shall we do?"
And Peter said to them, "Repent and be baptized every one of you in
the name of Jesus Christ for the forgiveness of your sins, and you
will receive the gift of the Holy Spirit. Acts 2:37-38 ESV
Reading this in English, it seems to indicate that Peter was instructing the people to both repent and be baptized for salvation, so they would need to do both. A Greek-speaking person who read this at that time would not likely draw the same conclusion, though, because the verb tenses of "Repent" and "be baptized" are different.
"Repent" is 2nd Person Plural (all of you) Active Imperative, while "be baptized" is 3rd Person Singular Passive Imperative. So, it doesn't appear to really say "All of you must repent and all of you must be baptized", but rather "All of you must repent and then let him (who has done so) be baptized." So, it seems that the translation into English has led to the uncertainty and misunderstanding. More Info
Jesus answered, "Truly, truly, I say to you, unless one is born of
water and the Spirit, he cannot enter the kingdom of God. 6 That which
is born of the flesh is flesh, and that which is born of the Spirit is
spirit. John 3:5-6 ESV
For this passage, the assumption is made that being "born of water" must necessarily be equivalent to being baptized in water. This is definitely one interpretation, but certainly not the only one. That assumption cannot just be "assumed" without a good reason to do so.
We must note that there are two times in which two things are joined by the "and"--one is water and spirit, and the other is the flesh and the Spirit: "born of flesh" and "born of the Spirit". Consequently, a valid interpretation is that water is referring to the flesh, as when someone is born the mother's water breaks. We also must recall that Nicodemus was tripped up thinking that he had to be born physically again and enter his mother's womb a second time. Jesus clarifies by indicating there are two distinct types of birth--a physical one and a spiritual one. Nicodemus had no need to be born physically a second time, but he did have need of being born spiritually.
The Scarcity of the Mention of Baptism
There are dozens of places where the word "believe" is used with regard to salvation where baptism is not mentioned at all. To hold to the position of baptism being required for salvation, one needs to give an explanation for this. If faith were insufficient for salvation apart from baptism in water, it is curious as to why it is so infrequently mentioned in these passages where faith is given as the means of salvation.
Why the Baptism of John rather than the Baptism of Jesus
John the Baptist stated very clearly that he was nothing compared to the Messiah who was coming. John baptized with water, but he spoke of One who would not baptize with water, but with the Holy Spirit.
I have baptized you with water, but he will baptize you with the Holy
Spirit. Mark 1:8 ESV
It is curious, then, that so often when the word baptism is seen in Scriptures that it is assumed to refer to John's baptism rather than that of Jesus.
So, the doctrine of baptismal regeneration comes mostly from interpretations of Acts 2:37-38 and John 3:5-6 that are a bit problematic.