Well, sure it could. But to me, if a quote from a person somehow reinforces a stereotype of a group the quoted person is labeled as part of, then that implies more about the person hearing the quote than the person stating the quote.

A person can use a quote out of context all day long but it doesn't truly reinforce the stereotype until a person hearing the quote agrees with the out-of-context usage.

For example, if a person from Mars says something stupid and someone provides me with a quote. Do I say to myself; "That person is stupid" or "Martians are stupid"? If I go with the second then the problem is within me, not the quote nor the person who provided it.

So, did the Nazi's quote Jews out of context to further their aims of denigrating them? I'm sure they did. But it only worked for people who agreed with the idea in the first place.

It should be if it were spoken that way, otherwise it's not accurate. Of course, that could depend on the context of the quote. If you are just communicating the information given then, sure, correct for grammar and whatnot. If it's to communicate the personality and/or attitude of the person, then you quote verbatim as it was said. That's why the term "sic" exists for quoting people, you are acknowledging that it is known the quoted statement appears incorrect. Because that's the way it was stated.