MorrisZapp from all I've read the history is clear that the argument was between the 'rights for all' approach of Anthony and the gradualist approach of Douglass, not a matter of either of them being 'against' the rights of the other group in question.
There are 3 quotes which are often brought up as evidence of Anthony's supposed racism. One is bogus (she never said it; it's a quote from a biography in which the author tries to express how she imagines SBA would have felt seeing voting expanded to Black men but not to women).
Here are the two that she did say, both in debate with Douglass over his request that she and other women suffragists divert their efforts from campaigning for votes for women, to instead support votes for Black men through the passage of the 15th Amendment:
"I will cut off this right arm of mine before I will ask for the ballot for the Negro and not for the woman."
and
“The old anti-slavery school says women must stand back and wait until the negroes shall be recognized. But we say, if you will not give the whole loaf of suffrage to the entire people, give it to the most intelligent first. If intelligence, justice, and morality are to have precedence in the government, let the question of the woman be brought up first and that of the negro last.”
That second one does make me wince, of course (the first one doesn't... the point she was making was in the and not for the woman part of the sentence).
The second quote sounds terrible out of context, somewhat less terrible in context. It was said during a public meeting where she and Douglass were debating whether the women's suffragist movement should mobilise in support of the 15th Amendment. Anthony isn't even arguing against the amendment; she's arguing against the position that women suffragists should temporarily tamp down their campaign for women's suffrage and expend their energies on helping pass that Amendment first. She was replying to Douglass, who had made some comments appealling for women's support by flattering them in that 'of course we all know that women are the more intelligent, refined and civilized sex; they can get their husbands to vote as they wish them to' [not a direct quote] kind of way. Anthony's response was a rhetorical device conveying something like 'well if you truly believe we're the superior sex, and you truly believe that fighting for one group at a time is the way to go about it, then why not pass women's suffrage first and see how happy you are with gradualism when it's your group that comes last?' The transcript records breaks with attendees laughing and applauding during both speakers' remarks, suggesting that the tone of the meeting was hard hitting but good natured on both parts. They were both extremely smart, eloquent people, and they were scoring points off each other on logic and rhetoric, not just moral righteousness.
It is true that some white supremacists used the fact that white women couldn't vote to argue against Black men being extended that right, saying it would put their own wives and mothers in an inferior position to those Black men, which they found unthinkable. Once Black men had got the legal right to vote, some within the women's suffragist movement felt they could exploit that perspective to gain support for womens suffrage, by making the case that now that 'Negroes' could vote, surely white men wouldn't deny their own wives and daughters the same level of citizenship? Some within the movement supported this approach as pragmatic, while others eschewed the approach of courting such unsavoury bedfellows and did not want to taint their righteous struggle with those associations.
As for Anthony, I don't think there's any evidence that she took that approach to the argument after the 15th Amendment was passed. Though even if she had, the view that "women shouldn't have fewer rights than Black men" is only offensive if you believe in Black liberation but not in Women's liberation.
Her record as an abolitionist and an advocate of universal suffrage can't really be disputed; she is on the record and the historical accounts of her political activity are there for anyome who wishes to look. It's only these two quotes (plus the false one) which get cited over and over by those who would dismiss her as racist.