I think they should have already been around in extremely small numbers with few incidents. The more powerful mutants have laid low and the majority of the very small number had powers that weren't very dangerous (like that one mutant who can make tattoos appear on people's skin). But the number of mutants has been increasing and their gifts becoming more powerful. This prompts Charles Xavier, who had been helping with world conflicts covertly using telepathy, to take action with the help of his human friend Dr. Moira MacTaggert to open his school to train young mutants to use their abilities. The status quo of mutants being relatively unknown with the X-Men as Xavier's secret response team lasts for about one movie until the unthinkable happens and a mutant teenager loses control of his powers in school killing and injuring a bunch of people. This leads the floodgates to open revealing mutants to the world and subjecting them to prejudice and spurring certain mutant extremists to action.
Now, here's the thing. Because Xavier's School is starting now, that means that the majority of mutants, including the X-Men . . . are teenagers. Which is in some ways fitting because it reflects an existing television show all about the X-Men existing in a world that doesn't know about mutants yet, ahem . . . https://www.youtube.com/watch?v=6faj24XtmP8. But I know that isn't everyone's cup of tea. Anyway, my personal line-up of choice is: Cyclops, Wolverine, Nightcrawler, Marvel Girl, Storm and Jubilee (yes, Wolverine as a teenager. We've already had the definitive regular version. So, now we can play around and have him as the "Judd Nelson from Breakfast Club" type character in this group). One other upside, having Storm show up in a world that doesn't know mutants but knows who Thor is means that we won't be painting Kenyan people as superstitious and backward if we include a certain part of her origin. Now, when a girl who can control rain and lightning shows up they'll just logically be like "Oh, I get it. She must be a god like that one you see on American TV. The Avengers have theirs and now we have one of our own".
Just one idea for how to do this. Honestly, I don't think the "X-Men Problem" is as much of a problem as people think it is. I mean, a lot of the questions people are bringing up are ones that people don't even bother asking about the comics. Like "If the X-Men were around, why weren't they helping". Answer: "I don't know. Where were the Avengers during all those X-Men crossovers in the comics? Somewhere else because it wasn't their comic book." And despite arguments that the movies need these questions answered because they're movies for the general public and have to be held to "a higher standard", I'm not sure that's true. If anything, I think they probably ask fewer questions than comic book fans.