Your question is one of the most ticklish ones to answer and every rabbi, especially progressive rabbis, have struggled with it.
However, I want to clarify something about your questions. We have not arrived at a ‘post-racial world.’ And, even if we had, Judaism cannot be defined as a race in the customary use of the word. I think what you were asking is that since our world – especially the Western world – is so integrated, why should intermarriage be such a big deal.
Indeed, for many people it isn’t a big deal at all. You can see this in every Jewish community – yes, even Orthodox ones, where a Jew is married to a non-Jew. However, in the Conservative and Reform communities, the intermarriage rate is high. This is especially true in different parts of the country where, for various reason of demography and culture, intermarriage rates can approach 70%.
But, to get back to your question. Why is intermarriage such a big deal? It is a question of covenant.
It is relatively easy for a child to be educated in a Jewish manner as to the meaning of covenant and being a part of the Jewish people if the parents are Jewish. Even if there are no children, where the religio-cultural bonds are unquestioned, connection to the Jewish world is easy. And that connection to the Jewish family and to the covenant between God and the Jewish people is what being Jewish is all about. Our faith is a brit – a promise to God that our Jewish family is bound to Him (pardon the gender specificity but grammar dictates it).
When Jews marry out of the faith, let me be clear that I do not believe they have committed a sin. But it takes much more work to create a Jewish home when one partner is not Jewish. I have seen many families succeed beautifully in this regard and their children are knowledgeable and good Jews dedicated to the Jewish people and to Jewish life. But the parents in these cases decided to raise their children Jewishly, have an exclusively Jewish home and live in a Jewish environment which they actively created.
On the other hand, there are many families that choose to live in at least two religious traditions. The children are raised with two traditions. I know of no rabbi who sanctions this. As well, we have seen families where the parents may be afraid to choose for the child and so decide to leave the issue on the table until the child can decide what religion s/he wants to be. Sometimes this works, often it does not for a variety of reasons.
The issue of interfaith marriages matters because marriage matters. Jewish families that want to identify as Jewish families and live as Jewish families actively bind themselves to Judaism and the Jewish people. Being a part of the covenant is an active step, not merely something you are born into. Diminishing its importance or value is a diminution of the Jewish people.