It seems to me that this stereotypical image of Elves as rather effeminate beings with long (often blonde) hair and high cheek bones has arisen primarily for two reasons.
First, in many children's tales (of relatively modern origin), Elves are depicted as diminutive, fairy-like creatures. An example would be the story of the Elves and the Shoemaker, but there are many other examples. And although Tolkien made clear in his writings that this was not how his Elves were to be viewed (and even expressed some regret at using the word "Elf"), many will still come to his works with preconceived notions of what Elves are. Unfortunately, his portrayal of the Rivendell Elves in The Hobbit doesn't exactly help his cause (
) and many readers first come to his works is by way of that book. But I think that it is fair to say that he was at pains to disabuse readers of such preconceived notions, particularly in the Silmarillion, but also in his depiction of the Elves that we meet in LotR (and even in the Mirkwood Elves later on in The Hobbit - Galion certainly seems to be a bit of a lad
).
But the second reason for this stereotypical image does, I think, lie in Tolkien's own description of his Elves. Notwithstanding his efforts to portray them as strong and majestic, he also (as has been said) conceived of them as a "fair folk". And it is difficult for those visualising his works (artists and film-makers alike) to depict them as such without borrowing from society's own notions of attractiveness. Which quite often means long flowing hair, delicate features and high cheekbones.
As Firefoot has quite rightly pointed out, this way of portraying Elves pre-dates the films. It is apparent in much of the artwork depicting Tolkien's works. And it is certainly apparent in the films. Whoever was responsible for casting certainly went for long blonde hair and high cheekbones when it came to casting the Elves, particularly the "chorus line" that turns up at Helm's Deep. Elrond is a notable exception, and, despite his boyish good looks, I would nevertheless maintain that Orlando Bloom's Legolas is no cissy.
Of course, it is Orlando who has become the butt of the "hairbrush and shampoo" jokes. But I have a feeling that this is not a new phenomenon. Such jokes are certainly more prevalent now, given the widespread popularity of the films (and accompanying resurgence in the popularity of the books), and also courtesy of the relatively recent technology which allows pictures (such as film stills) to be manipulated and distributed via the internet. But I seem to recall that the Legolas character in "Bored of the Rings" was rather obsessed with his appearance. And, when I drew a picture parodying the Fellowship some 20 or more years ago, I depicted Legolas with angular features, preening his hair and admiring himself in a compact mirror.
Both examples, of course, are a form of parody. As, indeed are the many jokes and images involving Legolas that are currently circulating. But I believe that the parody has its roots in the two basic reasons for the stereotype that I outlined above.