The premiere issue of Weird Tales was dated March 1923, with the byline “The Unique Magazine.” Within, editor Edwin Baird promised readers something different from the other magazines, but it was not immune from the general biases of its day. Filler articles for the first issue included quasi-Colonialist sentiments “Queer Tribe of Savages Found in Africa” and “African Brides Must be Plump”; the cover feature “Ooze” by Anthony M. Rud used terms like “darky” and “octoroon,” while Harold Ward apparently did not hesitate to use “nigger” in “The Skull.” In this, it did not differ very much from Adventure or other pulp magazines, where such terms were in common use—but stories in that inaugural issue of Weird Tales also focused on inherited insanity, “missing links,” and atavistic ape-men among other subjects of grim and grue, super-science and the supernatural.

Which is to say, most of the readership didn't find anything in Lovecraft's fiction worth commenting on. This is not to say that everyone agreed with everything that Lovecraft wrote, or that such disagreements were restricted to private correspondence - Charles D. Isaacson addressed Lovecraft's antisemitism and racism in Concerning the Conservative (1915).

Yet it is important to remember that while some of Lovecraft's prejudices strike readers today as blatantly racist, at the time and in the context of the pulps, his views were not unusual. Lovecraft by and large kept his weirder and stronger prejudices out of his fiction, and what you see in his fiction is fantasy racism - the Deep Ones in "The Shadow over Innsmouth" are undoubtedly inspired by then-current ideas of miscegenation, but the actual racism in the story is a red herring covering a much stranger reality:

But the real thing behind the way folks feel is simply race prejudice—and I don’t say I’m blaming those that hold it. I hate those Innsmouth folks myself, and I wouldn’t care to go to their town. I s’pose you know—though I can see you’re a Westerner by your talk—what a lot our New England ships used to have to do with queer ports in Africa, Asia, the South Seas, and everywhere else, and what queer kinds of people they sometimes brought back with ’em. You’ve probably heard about the Salem man that came home with a Chinese wife, and maybe you know there’s still a bunch of Fiji Islanders somewhere around Cape Cod.

The last part was true, by the way.

It is difficult to accurately gauge reader reactions because the letters columns of the pulp magazines were controlled by the editors; so we don't know if no one complained about racism in Lovecraft's stories, or if people complained but weren't published. Still, general fan reactions even in Lovecraft's private letters is positive, even from those who would challenge his prejudices. It's really difficult to get across exactly how racist the pulps could be at the time, but pulp fiction was often willing to pain very broadly with stereotypes, and racial caricatures featured very prominently in fiction from even the best writers of the era.