I am half-white English and half-Indian, and reasonably thick-skinned. But I can tell you that for the past sixty years, I have found that being referred to as "half caste" is really offensive, and strangely upsetting, distinct from all the other racial epithets in circulation.^^
I'm not sure why this particular phrase is so hurtful, but it is. Perhaps because it's so dehumanising. At least if you are called a Paki, or a Wog, there exists the implicit acknowledgment that you are at least a (type of) human being. But "half-caste", like "mulatatto" or "mixed breed" (an official British Empire designation, if you can believe that!) is like describing an animal or a thing.
In the USA, I have found that people with an ancestors from more than one European country, cheerfully refer to themselves as "mutts", often with a coy giggle. Of course in this context there is no power behind the word, and indeed it serves to let them not-so-subtly hammer home the fact that they are 100% white. You don't hear Americans of mixed racial background describe themselves as "mutts".
So, don't use the term "half-caste".
If it's absolutely crucial to you to nail down a precise description of person's racial pedigree, try "half-X and half-Y"
During all these decades, it's never occurred to me to use the formulation recommended by poster Sing, above: "both X and Y", which makes the point that you are a whole person, made of elements coming together. And if it's not biologically precise, too bad. Thanks, Sing.