The matter is worth considering. We ourselves grew up with "the New Testament in my blood, which is shed for you and for many, for the remission of sins." You know how it is with ritual incantations: the first learned are deepest rooted, and nothing else will ever sound quite right. (Which is why, for example, "debts" will never outpace "trespasses.") So let us confess up front that we are not without bias.
Nonetheless, there are problems with this translation. Some are easy -- the SBH put initial capitals on "New Testament." This can't possibly be right; the New Testament is a book, which when Jesus spoke had not yet been compiled. What the Lord offered in his blood was, rather, a new testament, comparable (albeit vastly superior) to the testaments (or covenants) with Noah and Abraham. The two are connected -- one describes the other -- but they are not identical. (The counter-argument is that we often capitalize "Word of God" when intending either the Scriptures or the Lord. That's another mistake, but a harder one to correct).
As for the words "for many," they are indeed an accurate literal translation of "pro multis" in the Editio Typica, and (more significantly) of "hyper pollon" in Mark 14:24. So it is easy to see why the traditionalists at NLM reveal so much indignation about the "erroneous" translation.
The problem is that translation is a complex art. Anyone who has ever tried to translate a few simple passages from Scripture has had to choose between literal and idiomatic renderings -- what are sometimes called "formal equivalency" and "dynamic equivalency." (Using formal equivalence, we would say in French that Dick Cheney is a batarde; but that would really be an aspersion upon his parentage which we are unprepared to support. What we mean to say is that he is a salaud.)
Kittel, our preferred guide to the complexities of Biblical Greek, argues at some length that polloi in the New Testament is generally used inclusively, and always so with reference to the saving work of Jesus. For example, in Mark 1:34, Jesus is said to heal "some" of the sick people around him; but both synoptic parallels amend this to read "all" of them, suggesting that this is how the developing tradition understood the Greek word. The argument proffered by Joachim Jeremias hinges on the connection of the key verses to Isaiah 53:12 (NRSV: "and he bore the sin of many"), and on the fact that Hebrew, lacking a precise verbal equivalent for "all" (in the sense of each individual; qol works for the totality of a thing), sometimes uses "many" to say "all."
Both translations, it seems to us, have a legitimate case. Beyond both, of course, lies the fundamental question of for whom, precisely, we be believe that Jesus shed his blood. All people? Most people? A chosen few? 144,000? This is a difficult place where one wants to shape doctrine according to the text, and yet finds oneself with little choice but to do the reverse. That said, and despite our nostalgic preference, we have to suggest that the blood of Christ was in fact shed for all people -- even those who reject it, and those (if any) who may not in the event derive any benefit from it.