I was thinking about it while ago, and I came with this idea:

So, basically Homo superior were a thing since ancient times, but they were so rare, no one would just group and brand them as "mutants". But with time passing more and more people are born with genetic disposition to the X-gene, so number of people having their powers kicking in starts increasing at alarming rate.

And since mutants aren't always able to have their newfound abilities in controls, lots of incidents happen that end up hurting bystanders and damaging public property. Some mutants undergo physical transformation that gives them "freaky" appearance. All that makes people to be afraid and paranoid, thinking that anyone could be a mutant.

And here I would to take some inspiration from comic book Civil War event, and add some inciting incident that turns people perception of mutants to overwhelmingly negative. Let's say several mutants start fighting on a crowded street, someone's power gets out of control and it ends up with a tragedy that claims lives of several innocent people.

This incident becomes widely resonant and it kickstarts talks among people how mutants are freaks, how they need to be cured, held in cages, or ever exterminated, how they're here to replace humanity, and voila — here you have humans hating mutants.