Challenge Your Own Views
Interracial dating is an inevitable consequence of a multi-racial society, and the arguments for accepting them have the same key point: Everybody is equal. It is impossible to predict someone's character based upon skin color, and members of any one race don't act the same in relationships. There will be people of your race who cheat or abuse their partner, but you wouldn't expect to be looked upon as liable to do the same just because you share the same skin tone. Don't look upon your son or daughter's partner in that way, either, because you wouldn't expect him or her to look at you in that way.
Looking for Love
It is a widely held prejudice that relationships between blacks and whites, for instance, are purely sexual. Again, think about it with regard to yourself. Would you be happy with a purely sexual relationship? If your son or daughter has introduced you to their partner, or is going to, it is unlikely that their relationship is purely sexual. They are trying to get you to accept them as a person.
It may help to talk to them. Don't ask directly, because this could sound threatening, but ask them about how they met, or whether their first date was awkward. You will soon find that their relationship is full of love and warmth. The same goes for a stereotypical notion such as blacks dating whites to increase their social standing. The odds of your family member's partner being a sociopath are as low as those of you yourself being one. If they are in a relationship, it is much more likely to be based on love.
If the relationship is across religious lines, it can be more difficult. However, you don't need to turn yourself over to their god; you just have to accept that they have. There are countless religions, and as long as you accept that no one can really know the answers to the big questions, you can also accept that some people do not agree with you. Chances are, if you talk to them, you will find that you share a lot of ideals.
Respecting Your Loved One's Decision
You should go to special effort to think of your family member. If you have any love for them, you should try to respect their decisions. Although you are trying to protect them from being hurt, they are free to make their own choices in life. Hurt can be an unfortunate consequence of that, but lasting happiness is the other possibility. Your family member's partner is an individual, and they should be treated as such. Someone you are very close to, or even possibly raised, has reserved a place in their heart for this person. This makes it likely that if you talk to them, you will see something you like in them too.