What is the biblical basis against baptismal regeneration? Several different denominations teach that water baptism is a prerequisite for (or is essential for) salvation. What is the biblical basis against this teaching?
One of the best articles on this controversial subject is from the Christian Apologetics and Research Ministry, entitled Is Baptism Necessary for Salvation? by Matt Slick. I've included some of the key verses it refers to, and arguments it uses, below.

Romans 5:1 and Ephesians 2:8 show that we are justified by faith (not by faith + something else). Furthermore, Romans 4:1-11 talks of being saved by faith, not by works - including the "works" of the Jewish religious ceremony of circumcision. In particular, verses 9b-10:


  We have been saying that Abraham’s faith was credited to him as
  righteousness. Under what circumstances was it credited? Was it
  after he was circumcised, or before? It was not after, but before!


In other words, Abraham's faith saved him before his circumcision; salvation comes before the religious ceremony, not as a result of it. 

As a slight aside, another oft-cited example is the account of Jesus' crucifixion (Luke 23:40-43) where one of the criminals being crucified alongside Jesus is apparently saved without being baptised. Slick does not quote this, but does make mention of the many people who receive Christ on a hospital deathbed and don't get baptised before they die.

Colossians 2:11-12 links the Old Testament ceremony of circumcision with the New Testament ceremony of baptism; therefore what applied to circumcision (it follows salvation, which is by faith) applies to baptism.

1 Corinthians 15:1-5 says that it is by the Gospel that we are saved, and defines the Gospel as Christ dying for our sins and rising again. We are saved by Christ's endeavour, not our own. Paul emphasizes this earlier in 1 Corinthians 1:17a: "For Christ did not send me to baptize, but to preach the gospel" - clearly the two are different things, and it is the latter that Paul sees as the more important.

There are several other passages that say "if X then you are saved" where X does not include baptism; Romans 10:9 is a good one:


  If you declare with your mouth, “Jesus is Lord,” and believe in your
  heart that God raised him from the dead, you will be saved.


Equally, Romans 6:23 describes eternal life in Christ as a free gift - in other words, we do not have to pay, or undergo any religious rite, in order to receive it.

Slick's article then goes on to explain those verses that may seem to say that baptism is part of salvation. That's beyond the scope of this question but I would recommend reading it when looking at this issue, particularly when considering the flip-side of this coin, the question asking for the Biblical basis for baptism as a prerequisite for salvation.