I'll begin by asserting that there is overwhelming reason to conclude that the New Testament was reliably transmitted from the original writings to the copies (the Greek manuscripts) that still exist. I will be quick to acknowledge, however, that the thousands of manuscripts available to us do contain variants, places where they disagree with one another. Indeed, there are literally tens of thousands of such variants among these copies. That being the case, isn't the charge of tainted transmission a valid one?
No, not at all. For one thing, the vast majority of these variants are completely insignificant. They amount to nothing more than an alternate spelling or the fact that a single place or person was known by two different names. So the issue of the reliability of the copying comes down to approximately 2,000 places where variant readings that are not insignificant can be found among the manuscripts. Most of these will be identified (by footnotes or marginal notes) in any good study Bible.
Let me share two examples of significant variants, one accidental and one likely intentional. In Romans 5:1, the Greek word εχομεν or εχωμεν appears in the different ancient copies. The difference is the third letter--did the original contain an omega or an omicron (the two different Greek 'o's)? In English, the verse reads
Therefore, since we have been justified by faith, we have peace with God through our Lord Jesus Christ.Or..
Therefore, since we have been justified by faith, let us have peace with God through our Lord Jesus Christ.Is Paul telling us that peace is an accomplished fact or something that we should be striving to appropriate? In this case we have a very minor alteration that leads to a rather significant difference in meaning. The incorrect insertion of the wrong 'o' would have been an easy mistake to make, especially if the scribe were listening to someone else dictate the letter.
There's an example of an intentional error in the second verse of Mark's gospel. Some copies read, "As it is written in Isaiah the prophet..." This is followed by an Old Testament quote, a quote which can be found not only in Isaiah but also in Malachi. So other copies read, "As it is written in the prophets..." It would seem that some first or second century scribe, in copying Mark's written account of the events of Jesus' life, decided that Mark hadn't been on his game when he wrote "in Isaiah the prophet." The scribe 'improved' the text by changing it to "the prophets." Both variants, of course, are correct, but the first is likely what Mark actually wrote.
There is, therefore, an entire field of scholarship called New Testament textual criticism that seeks to recover the autographs by careful scrutiny of the wealth of copies in existence. Scholars in this field examine external evidence (including the dates and locations of the variants in the Greek manuscripts, in the early Latin, Coptic, and Syriac translations, and in the citations from the early church fathers) and internal evidence (such as 'which variant best explains how the other arose?'). The result is a level of certainty about the originals that exceeds 99% accuracy.
It is important to note that no Christian doctrine is undermined by any of the variant readings. If we were to ignore all of the passages in which variants are found--and use only those passages in which all the relevant copies agree completely--what would be the result? We would have the very same picture of Jesus--a miracle-working, divine Son of God who died by crucifixion and three days later was raised in a glorified physical body.
The existence of errors in copying--some of them significant--should cause us no concern with regard to the reliability of the New Testament. We do need to recognize, though, that the referent of the biblical doctrine of inspiration (and of the implied doctrine of inerrancy) is not a particular set of copies--much less a particular English translation--but the autographs. These we don't have, but--through the reasoned application of New Testament textual criticism--we have a great deal of certainty about what these originals contained.