Wikipedia has the origin I've heard:
The word "beatnik" was coined by Herb Caen in an article in the San Francisco Chronicle on April 2, 1958. Caen coined the term by adding the Russian suffix -nik after Sputnik I to the Beat Generation. Caen's column with the word came six months after the launch of Sputnik. Objecting to Caen's twist on the term, Allen Ginsberg wrote to the New York Times to deplore "the foul word beatnik," commenting, "If beatniks and not illuminated Beat poets overrun this country, they will have been created not by Kerouac but by industries of mass communication which continue to brainwash man."
But I love Daniel's examination of "betelnut"!
And a further comment from Daniel:
For an alternate claim about the origin of “beatnik”, search the WWWeb for
"Sam Katzman" beatnik
It should be noted that, while the Katzman story and the Caen story are often told with reference to “Sputnik”, both Katzman and Caen would have been familiar with the Yiddish “-nik”, meaning one who is in some way characterized by the prefixing term (as in “refuse-nik”, one refused the right to emigrate).