When I told my friends on the East Coast that I was moving out to California for grad school, many of them responded with comments like, "Please move back east before California falls into the ocean," or, "Be sure to buy a life vest once you're out there!" When I explained in turn that there was no such danger, illustrated with highly-technical sliding of hands past each other to explain the term "strike slip," most of those friends then told me that I was taking them way too seriously on their comments. One, however, still told me not to jump on the San Andreas Fault, just in case.
The fact that this came up in casual conversation with so many people, even on the East Coast, is only a small sign of how pervasive this misconception is. It's become a part of pop culture even more than it's something people actually worry about - well, I certainly hope it's more pop culture than genuine belief, but that would be a hard thing to gauge, and my circle of nerdy friends is probably not the best place to calculate such things. A quick Google search for "California fall into ocean" reveals all too many sites, some of which are those user-asks-question-another-user-answers database sorts of things, which assert that a big splashdown is imminent (this makes one wonder about the credibility of the other information on those question sites). And friend of mine did tell me recently that one of his relatives moved out of California after Northridge and bought a map with the supposed new West Coastline of the United States. One also has to wonder if that map was drawn as a joke, or if the mapmaker is hoping to use those funds from the maps to move away from the supposed sinking slab.
In some ways, the concept of the entirety of California going down doesn't seem too far fetched. This is, after all, a state that routinely has hundreds of square miles catch on fire, where even a moderate amount of rain sends people out sandbagging to curb any neighborhood-demolishing debris flows, where even not-The-Big-One can pancake high rises every decade or so. Not to mention those extremely-poorly-placed beach clifftop houses that do tend to surf the mass wasting events and wipe out in the water. It doesn't seem strange at all that an active imagination used to seeing such stories on the news might extrapolate to an even larger-scale disaster. No matter how much that thought may worry the imaginer, it only becomes truly troublesome when the actual science that disproves the myth gets ignored.
The California falling into the ocean myth actually predates much of the science that disproves it. Though I haven't been able to find any specific date or single source for the origin of this misconception, I have seen references to it in relation to the 1906 San Francisco quake. Following that temblor, all sorts of exaggerated "news" stories wired across the country, including one that the strong shaking had brought the ground under San Francisco down along with the city's buildings; this story quickly expanded to encompass all of California going under. When the San Andreas Fault was eventually blamed for the temblor and mapped out as extending through most of the state, the potential for more earthquakes and the clear line of breakage could have only fed the thought among the non-scientifically-inclined.
(This leads to a sub-misconception. Every version of the story I've seen cites the San Andreas as the line on which the break will occur, but many seem, well, shaky on where the Fault actually is. Some put it as the geographical line that separates California from Nevada. A great many other retellings of the story say that Los Angeles in particular will be swimming with the fishes, with no particular mention of San Francisco, San Diego, or especially not San Bernardino going along with it.)
The best way to fend this one off is, I think, better geology classes earlier on in school. None of this waiting until 12th grade to take a senior “slacker class” business – that’s more than enough time for the conspiracy theorists to get to someone. I don’t think any basic part of this misconception is too complicated to explain to someone in, say, middle school, if not even earlier.
But for all I hope people stop actually believing that California really is going to break off at the San Andreas Fault during a major earthquake and fall into the ocean, I would be sad if the idea faded away from the silly pop culture side of things. Without this myth, there wouldn’t have been those gloriously terrible sequences at the climax of the 1978 Superman movie, in which Lex Luthor implodes the Carrizo Plain section of the fault (highlighting another Faults Don’t Work That Way myth), then Superman flies inside the magma-ridden(!) crevasse to fix it, prompting the exact same footage of the collapse to be shown in reverse. Nor could that popular “End of the World” flash cartoon (NSFW for four-letter words) have gone quite as far with the silliness as it did. And this story has also prompted its fair share of songs, from Tool’s angry-good-riddance "Aenima" (also language NSFW), to Cass Elliott’s happy-good-riddance "California Earthquake", and the more obscure LA-based Little Girls’ perfect parody on SoCal surf tune style "Earthquake Song". Once people can recognize misconceptions as nothing more than that, why not have fun with them?