I think it's possible. There's a certain amount of power from faith and belief in the Dresdenverse. It's enough to supercharge the phony Shroud of Turin, for example (though the real one makes that look like a C battery). If enough people knew about it and believed in it, it might be enough to strengthen the Things that are out there.
The supernatural in the Dresdenverse doesn't seem to be as driven by belief as a lot of other books (like "American Gods," for example), but it definitely has something. Magic doesn't work if you don't think it will (or should), and Harry speculates that Mab hooked up Walt Disney with something extra to keep fairy tales in the public's mind, for example.
I do think that the mortal world will know before the end, though. In fact, it might be the End. In my head, Humanity ends up taking the reins after the Big Apocalyptic Trilogy. The Paranet is kind of a microcosm of what vanilla mortals could do; one nerd in his mom's basement gathers and correlates enough data to determine that something weird is going on in Lake Michigan in Cold Days. Humanity is more than capable of gathering enough information from seemingly unimportant data sources. It's only a matter of time before the supernatural version of WikiLeaks dumps a ton of data about the spooky side of things in the Dresdenverse (in my opinion). You can't keep stuff hidden forever, not anymore; data no longer decays, and it is no longer stored centrally. That shady government force that "disappeared" the Fool Moon tape wouldn't have been able to do it if it was digital. Hell, I'd be shocked if there weren't copies out there even in the Dresdenverse; there are plenty of OCD people who probably taped the news on VHS regularly.
And Digital copies are no longer distinguishable from the source. That's why so many watermark their images now. Used to be that photocopies and ditto machines were the best we could do; now we've got one copy that can be replicated an infinite number of times without degradation, all accessible from anywhere in the world, at any time.
So, yeah. I think the mortal world will know. They'd have to be sublimely stupid not to.