Y'got some... logical problems going on, there.

Firstly, your (1), while true, is misleading; it should read "If you believe that subjective opinions which are not based on evidence are morally acceptable, then you must believe that [long, LONG, probably literally infinite list of possible views, of which sexism and racism may be members but which contains innumerably more other stuff] are morally acceptable". Sure, accepting beliefs without evidence may lead us to sexism and/or racism, but that's hardly our biggest problem at that point.

Secondly, you presuppose that sexism and racism are necessarily not based on evidence. Of course, you may say that sexism and racism are by definition not based on evidence, because if there's evidence, then it's not sexist/racist, but that would be one of those "37 Ways That Bad Stuff Can Happen" or what have you; most people, after all, do not use your definition of "sexist" or "racist"; the common definition takes no notice of whether there's evidence or not.

Thirdly, for every modus ponens there is a modus tollens — and, as in this case, vice versa: we could decide that "subjective" opinions not based on evidence are morally acceptable (after all, we're not talking about empirical matters, right? These are moral positions). This, by your (1) and modus ponens, would lead us to accept sexism and racism. Intended? Or no?

Finally — and this is the big one — it strikes me as fundamentally backwards to start from broad moral positions, and reason from them to a decision about whether we need evidence for our moral positions.