One way I judge such things is how it impacts my life. I grew up in the Midwest, where the population is predominantly white. My father, in particular, grew up in a home which was not overtly racist, but he certainly had some racist attitudes--primarily because he had so little contact with people of color. If I had chosen to date or marry a person of another race back in 1980, my father would not have been supportive. If I were to do so today, I think he'd be at least open to the concept. It seems to me every generation will be more tolerant of the concept. Some people and communities have "arrived" and are accepting of such things; others certainly have a way to go.
The first response was quite helpful, but I want to give you a bit of legal history, too. Until the United States Supreme Court issued its ruling in a case called Loving versus Virginia, there were laws on the books making it illegal for black and white people to marry. I have provided a link, since you might find it interesting to read the court opinion for yourself. What is quite striking is that this opinion was not handed down until 1967.
Now, since it was illegal in some states for people to marry outside of their race, you can also see that their offspring did not fare all that well, either. But this varied, depending on the racial mixture. For example, there were many soldiers who brought home brides from Korea and Japan, and their children never experienced the prejudice experienced by the children of black and white marriages. Many people have married Native Americans, and while there was not as much prejudice as there was against a black and white mix, there was more prejudice, I believe, than there was against a Far Asian and white mix.
Given that backdrop, I do think that the situation has improved, but the degree of acceptance is often a function of the community one lives in, with some communities being much better than others.
In the 1940's the idea of a black man and a white woman was all but considered a crime. In the south, southerners still viewed it as a crime and would often take matters into their own hands and lynch any black man who made the slightest attempt to hook up with a white woman would be beaten or lunched.
Even in the north, the relationships between whites and blacks were considered taboo. Most black men were more accepting of the relationship, but even black women often had a hard time accepting the relationships.
Our country has come along way in changing whether relationships among races is concerned. Even over the last twenty years they have become more acceptable. Bi-racial children are no longer the abnormal child in society. There are many bi-racial children and multi-racial children in the school systems and country.
Prejudice still exists and there are always going to be varying degrees of prejudice, but the world has become more closed in and as it does, our society finds more interracial marriage alliances.