*Exactly who "they" [who say it] are, in this case as in many others, is unclear. My mother used to say it before I started school in September 1956. The phrase is often attributed to George Ball, a US Undersecretary of State in JFK's and LBJ's administrations, but his first recorded usage of it is in the 1960s. Ian Fleming used it as a chapter title in Diamonds are forever, published (London: Jonathan Cape) in 1956; but I am fairly certain that my mother has never read any Ian Fleming, and Fleming seems, in any case, to be quoting.
Just so. Ball is widely quoted as using the aphorism - later called the "Ball rule of power" - concerning the political value of closeness to the President (Walter Mondale is also credited with its use in that context in the mid-1970s). Fleming, in the chapter of the same name in Diamonds are Forever, put the words into the mouth of Felix Leiter, as he says goodbye to the still-bickering Bond and Tiffany Case.
"And you've got your plane to catch. You can go on fighting at twenty thousand feet. Get a better perspective from there. May even decide to make up and be friends. You know how they say." He beckoned to the waiter. "Nothing propinks like propinquity."
But where did it come from before that? A number of unspecific citations - the spelling varies ("propinks", "propinqs", "propinques") - call it "an old saying", "a modern saying", and so on. A couple specifically credit PG Wodehouse (Sarah Bradford, 1984, and Petronella Wyatt, 1999) but although Wodehouse clearly relishes the word (see Right-Ho, Jeeves) there's no sign of this specific quotation, so it looks like a "Hillfinger". The same goes for Frank Welsh's 1982 attribution to "the immortal words of Groucho Marx".
There are a number of precursors, "nothing ... propinquity" sayings to exactly the same effect:
There is nothing like propinquity to intensify friendship
- New Outlook, Alfred Emanuel Smith, 1918
Nothing breeds interest like propinquity
- A Heroine of Reality: A Novel, Percy Vincent Donovan, 1903
"nothing fosters a passion like propinquity"
- Arthur's Home Magazine, 1892
As Miss Edgeworth says, Cupid desires nought so favourable as propinquity
- A Midshipman in Love, Harvardiana, Vol III, No II, October 1836
"Miss Edgeworth" is an allusion to the novelist Maria Edgeworth (1767-1849) who repeatedly stressed the role of "propinquity" in cementing relationships (as in the 1809 Tales of Fashionable Life - see quotation), and that seems to be the root of the saying. But there's no sign pre-Fleming of the specific form, so he wins the Internets for the first solid citation. However, it'd be nice to find the coiner of propinking.