Can "mongrel" be used to refer to people?

As @Green Grasso Holm said in her answer, mongrel can be used for a person. But, as many others commented, it is an extremely offensive term when applied to a person. (And it is an insult to many mixed breed dogs, who tend to be every bit as intelligent and fine in character as purebreds.)

As long as you make it absolutely clear that this is a offensive term used by a racist society or by the bad guys in your story, you may get away with it.

I wondered if Mr. Spock, half human and half Vulcan, was ever called a mongrel in Star Trek, so I asked on Science Fiction and Fantasy Stack Exchange: Was Spock ever called a mongrel in a Star Trek TV episode or movie?

The answer by @Buzz is No, although Spock was called a half-breed in three episodes. According to a comment by @wcullen, Spock was referred to as a mongrel in a non-canon novel Shocks of Adversity, but non-canon doesn't count among Star Trek aficionados. Finally, there was an interview with Leonard Nimoy, who played Spock, in which he calls Spock a mongrel, but the wording is ambiguous and Nimoy may possibly be saying that Vulcans regarded Spock as a mongrel. Or not.

I've gone into all this detail on Spock because he is the most famous mixed-species person in all of science fiction, perhaps in all literature, and standards were much more lax when Star Trek came out. So the conclusion is:

As far as the English goes, you can use it, but be careful if you do use it.