Any influence there was unconscious and loose.
Stan came up with the idea of mutants to avoid having to come up with unique superpower origins for every character. Their powers are the result of random genetic mutation. They are born that way, although their powers don't usually manifest until adolescence.
Many humans would naturally fear the power that these mutants wield. Some mutants will misuse their powers. It's a natural source of conflict and a natural metaphor for power, bullying, bigotry, and civil rights issues.
Woke activists point to this as proof that the X-Men comics were always woke. They make similar claims with other media and franchises, like Star Trek. But they either deliberately or ignorantly confuse liberalism or progressivism with wokeism. They conflate two very different strains of political thought. Wokeism is fundamentally illiberal and Neo-Marxist. MLK's dream of a colorblind society is now considered to be dangerous and racist by woke activists. They have more in common with the mutant supremacist terrorist Magneto than the idealistic Professor X.
Stan himself has said "on social issues, I try to get it in in the background or underlaying the plot, but never to the point of letting it interfere with the story or hitting the reader over the head." This is contrary to the modern woke mode of storytelling, which tends to be highly partisan, overt, heavy-handed, and propagandistic.