The answer posted by Jack O'Flaherty seems quite plausible in the context of the cited movie script. Nevertheless, the only instances of "white ribbon" that I've been able to find in slang dictionaries identifies the term as referring to gin.
For example, from J.S. Farmer & W.E. Henley, Slang and Its Analogues Past and Present (1904):
WHITE-SATIN (-LACE, -TAPE, -WINE, or -RIBBON), subs. phr. (common). — Gin [cross-references omitted]
From Eric Partridge, A Dictionary of Slang and Unconventional English, second edition (1938):
white ribbon, satin, tape, wine, wool ; also w[hite] lace. Gin : low : resp[ectively] C 19–20 ; from ca. 1720 (A New Canting Dict[ionary], 1725); 1820 (Randall's Diary); from ca. 1780 (Grose, 1st ed.); mid-C 19–20,— occ[asionally] merely lace or its synonym driz. [J.C.] H[otten, The Slang Dictionary], 1st ed., describes w. satin and w. tape as women's terms, as, also, was lace. All are ob[solete] ; in fact, white wine and w. wool did not survive beyond C. 19; white satin may well endure, however, because of the trade name White Satin Gin.
As suggested by Partridge, J.C. Hotten, The Slang Dictionary: Etymological, Historical and Anecdotal, new edition (1874): has these related entries:
White satin, gin,—term amongst women. See SATIN ["Satin, gin ; 'a yard of SATIN,' a glass of gin. Term used by females on make-believe errands, when the real object of their departure from the home to replenish the private bottle. With servants the words 'tape' and 'ribbon' are more common, the purchase of these feminine requirements being the general excuse for asking to 'run out for a little while.'"]
White tape, gin, —term used principally by female servants. See RIBBON ["Ribbon, gin, or other spirits. Modification of white satin"].
White wine, the fashionable term for gin. [Example:] "Jack Randall then impatient rose, / And said, 'Tom's speech were just as fine / If he would call that first of GOES / By that genteeler name—WHITE WINE.'" —Randall's Diary, 1820
And from Jonathon Green, Cassell's Dictionary of Slang, second edition ((2005):
white ribbon n. (also white ribbin) {early 19C} gin.
Green notes that in the early to middle nineteenth century ribbon (by itself) might refer to "gin; spirits in general {var. on SATIN and like it implying the smoothness of good gin}."
All of these instances of "white ribbon" (etc.) as slang for gin are from British sources. The adoption of the white ribbon as a symbol of total abstinence from alcohol and other pernicious societal temptations seems to have originated in the United States, under the auspices of the Woman's Christian Temperance Union, although it quickly spread to Britain as well. In fact, the earliest published mention of "White Ribbon movement" that a Google Books search turns up is from a London periodical, The Bible Christian Magazine (May[?] 1883):
TIVERTON.—Our Quarterly Meeting was held at Cheriton Fitzpaine, March 26th. ... A public tea was provided, of which about 100 persons partook. A public meeting was held in connection with the White Ribbon movement, which has been organized here about a month, having already 85 m4mbers enrolled.
Obviously, "white ribbon" had a very different meaning within the WCTU of the late nineteenth century than on the streets of early nineteenth-century London. Frances Willard, "Temperance Legislation, Past and Future," in The Review of Churches (February 1893) explains:
The White Ribbon movement, so called from its badge, the symbol of peace, purity, and philanthropy, has its headquarters in Chicago, where the Women's Temple has been built a source of revenue to the Society from rentals, besides being its official headquarters. Here is its publishing house (founded and conducted wholly by women), here it publishes three papers, and sends out over a hundred million pages of Temperance and Purity literature in a year. The Society has its own "Lecture Bureau" for furnishing speakers and organizers; its own Press department for supplying facts, arguments, and news to the press. A National Temperance Hospital and Training School for Nurses have been established in another part of the city, to demonstrate the practicability of the non-use of alcoholics in medicine.
B.O. Flowers, "Some of Civilization's Silent Currents," in The Arena (November 1892) offers a vivid picture of the White Ribbon movement and other contemporaneous improvement societies:
The Woman's Christian Temperance Union and its sister organization, the White Ribbon movement for social purity, are being felt more or less in millions of homes. The White Cross movement; societies for home culture and for ethical training; summer schools of philosophy, science and ethics; college extension; and Associated Charities, with all the encumbrances of conventionalism, are leavening society and doing far more than we realize to keep in check the baleful influences of the saloon, of the incoming tide of ignorant and vicious emigration, of the aggressive democracy of crime and vice now within our borders, and the vicious spirit of the business world, which to so large an extent worships gold and loses all finer thought in thought of self.
Although it is tempting to imagine early prohibitionists commandeering the British servant-girl slang term for gin and transforming it into a symbol of what Flowers calls "social purity" and Willard calls "peace, purity, and philanthropy," it seems far more likely that the older euphemistic "white ribbon" of make-believe errands has no connection whatsoever to the newer "white ribbon" of moral uplift, social purity, and alcohol-free medicine. An embrace of tee-total abstinence from alcohol—if not of the other pillars of the WCTU's White Ribbon movement—does indeed seem to be the sense of the phrase "went White Ribbon" as used in the 1947 film Nightmare Alley.