I used Google Translate to say "I'm not Korean."
I expected something like:
However, I got this:
Why is the で used in this situation?
'to be (a thing)' is a bit of an odd complex verb thing, involving both で and ある. Your first sentence, lacking で, simply means 'I have no (inanimate) Koreans'. The negative is, a bit unusually, ではない in most cases rather than just でない, but the で cannot be omitted - it's part of the verb である(～だ), and without it, all you have is ある (which is only 'for an inanimate thing to exist or be located').
If you mean you expected "I'm not Korean." to be written 私は韓国人はない, then it seems you forgot that it's not は, but じゃ.
There are several ways of writing this simple negative statement, of varying degrees of politeness/formality.
Someone with a better grasp of grammar can probably tell you what function で plays in this construction...
Actually I found a previous Q&A that applies and goes into slightly more detail in the link below: