Star Trek is a product of American culture, and like almost all narratives produced by that culture, some of its subtext reflects racist ideologies that pervade America even if subconsciously -- such as the assumption of monocultural hegemony throughout entire species, or the idea that genetics really can equal destiny in terms of personality and abilities. The resemblance some alien species and their cultural practices have to common white supremacist stereotypes of non-whites, in particular, is a fair criticism. (The original design for the Klingons on TOS is particularly noteworthy -- they come dangerously close to a Fu Manchu-esque "Yellow Peril" stereotype of Chinese culture; the use of dark skin, wide noses, and association of Klingons with seemingly irrational violence later prompted some critics to wonder if the TNG-era Klingon design reflect subconscious white stereotypes of African Americans. And don't get me started on the Kazon -- originally developed as the "Bloods and the Crips in Space," the entire creative conceit behind the Kazon as a fictional culture is troubling in its racial attitudes towards the role of street gangs in the inner city.)
On the other hand, Star Trek was consciously designed to argue against overt forms of racism, as understood from a mainstream Kennedy Democratic perspective at the time. That perspective is often lacking from a contemporary point of view -- it's all well and good to say that everyone's equal, but Star Trek still only ever shows the white guy as the Hero In Charge and the non-white guys as his fawning subordinates -- but it also ain't nothin'. Sulu may have been the driver and Uhura the receptionist IN SPAAACE, but by the same token, there were no Stepin Fetchits on Star Trek. We saw Kirk answer to admirals who were black and Latino, and overt racist ideology was condemned in numerous episodes. (Really, the pervasive sexism of TOS is, I think, far more pernicious than its unconscious racism.)
Star Trek has also gotten better at this over the years. Deep Space Nine, in particular, did a lot both to develop female characters that were complex and had agency, and to both demonstrate egalitarian cultural diversity among humans and to deconstruct the "Planet of Hats" monoculture practice for its aliens -- Ferenginar was full of low-level conflicts between the patriarchal capitalist establishment and a feminist social democratic movement; Cardassia was divided between military imperialists and democratic reformers; Bajor had numerous different factions with conflicting religious beliefs about the Prophets and the Pagh-wraiths; the Klingon Empire was divided by true believers in Kahless's teachings and others who were more opportunistic in their beliefs; even the Dominion was divided in some ways between some Jem'Hadar who believed in genuine selfless devotion to the Founders, and some Vorta with a more cynical attitude. Deep Space Nine didn't take any of this as far as it could go, but it added a layer of depth and complexity to the alien cultures of Star Trek which earlier (and later) shows often lacked. Star Trek: Enterprise's later depiction of Vulcan cultural diversity -- melders vs. mainstream Vulcan society, Vulcans who rejected emotion suppression, Syrannites vs. the High Command, Vulcan democrats vs. Vulcan authoritarians like V'Las -- owes a lot to the work Deep Space Nine did to develop alien cultures as heterogeneous entities.
I am looking forward to Star Trek: Discovery. I'm glad Fuller and the subsequent team took the idea of having a genuinely diverse cast to heart, and I hope DSC continues the work of overtly fighting racism and of deconstructing subconsciously racist subtext, and the same work for patriarchy and heterosexism.
Oh, and those knuckle-draggers complaining about "SJWs are ruining Star Trek" and saying that not making white guys the center of the universe = white genocide? Fuck that noise and fuck those guys. (Hell, those alt-right Trumpists probably would hate the original Star Trek if it had premiered today -- after all, it starred a Jewish Canadian immigrant and a Jewish American son of Ukrainian immigrants.)