Like many things in life, this will obviously depend on the specific situation. For example, if the relevant people understand your lifestyle and why you would be sensitive to this issue before it came up would be a very different question than if they are militantly opposed to your zealous bigotry.
I had a close relative marry a non-Jew, and I actually had one party (the immediate family of the person getting married) who appreciated my lifestyle choices, understood I meant no offense, and needed very little explanation other than answering the question "you won't be able to come, will you?" and another party (other relatives) who were mortally offended that I wouldn't come.
I tried two things - the more effective point which I made was that as someone who many of my non-religious relatives looked to as a representatives of religious Judaism, and whether or not I would like to attend, I couldn't give off the impression of condoning something which is against my belief system. The less effective argument that I made (although I don't know if it was worthless) was that they can't appreciate what this issue means to me because they see it as a matter of personal preference, and why should my preferences mean offending someone else. I explained that to me, this is something objectively not okay, and just as they wouldn't ask me to attend a murder, they shouldn't ask me to attend this event. (In retrospect, the murder example was a bad one - despite the fact that it is logically a good point, people don't like what they are doing to be compared to murder.)
With one particularly vociferous relative, I pointed out that he was accusing me of being close minded while he was guilty of being close minded against my prerogative to my own beliefs - I wasn't trying to cancel the wedding, I was just choosing not to participate. If they have the right to make the wedding, I have the right to not join in. I was very surprised that this was well-taken.
Like I said, everyone's family and situation is different, and this is what (seemingly) worked for me.