I read (and try to write) SF and F, and a number of respected authors in those genres published short stories before they did a novel. I don't know how the actual percentages break down, as I certainly know of some who are only novelists, or who didn't publish short stories until selling their first novel or so. The biggest awards in the genre--Huge, Nebula etc. have categories for short fiction.
I think it's probably true that no one makes a living, or much more than pocket change, as a writer of short fiction alone in SF and F, and the SFF writers who are household words these days are best known for their novelists. However, it's also true that most published novelists in the genre don't support themselves with writing alone either. As I understand it, the current market doesn't allow most mid listers to live on the money from writing alone, and even writers whose books are "lead titles" when they come out can struggle to earn enough, unless they're very prolific, or their books become breakaway bestsellers. The issues that accompany taking the plunge to being a full-time writer has been a subject some have blogged about.
I guess my question for the OP is do you read a lot of short fiction in your favorite genre? If so, where do you obtain your short stories and how much are you willing to pay for access to magazines, short story collections, or online access to such? Would you pay for access to a stand-alone short story (sold on Amazon or somewhere) by an author you don't already know well, and if so, how much?
I'm guessing it's always been hard to support oneself on short fiction alone, unless one is really famous and their stories fetch prices that are far beyond market rates. Modern "pro rate" SFF markets (according to the membership qualifications for the SFWA) pay six to ten cents a word (most are six cents or just above). I don't know how this stacks up with other genres of short fiction, but it seems like there's a limit to how much a publication can pay per word without becoming prohibitively expensive and losing its readership.
Doing the math, this means that the highest price a 5000-word SF or F short story would fetch is 500.000 if the author sold to a top-rate market (and not many pay ten cents a word in SFF, at least, and they're brutally competitive markets). A novelette or novella-length work could be more lucrative, but you'd still have to sell a huge number of stories to top markets to make, say, 50k a year, or whatever is regarded as a middle-of-the road income. There aren't that many short fiction markets to sub to, at least not in SFF (which still has the rep for having a pretty active and robust market for short fiction).