"The old west" is a bit vague as a category, but I'll speak generally about frontier towns, particularly mining towns, and give some data from Western United States in the 19th century. Mining boomtowns were overwhelmingly male, few women were present; in an place like a drinking establishment, it would be unexpected to find a woman not involved in the entertaining of men in some way.

Note that frontier and mining towns around the world share a lot in common, and we can examine these environments closely in contemporary settings like the Amazon:

On Highway PA-275 near the 1980 gold rush boomtown Curionopolis in the eastern Brazilian Amazon one passes a fading hand painted sign reading: "OLD WEST RANCH." Arriving in the next town Parauapebas by this the only highway to the rest of the country, the first side road one encounters is the "Street of Happiness," where during the peak of construction of the giant Carajas mines between 1981 and 1985 about 250 brothels gave the neighborhood its boisterous reputation. The town's largest bordello, the "Pinga de Ouro" (Drop of Gold), faced directly on the main highway and had in one long barn-like shed fifty-seven wooden stalls. At the peak of the construction phase of Carajas, in the early 1980s 25,000 men were employed in temporary jobs.

The estimated adult female population of the town at the time was 1500, of whom perhaps 500 were estimated to be prostitutes. This doesn't suggest a comfortable environment for casual social drinking by women, and it isn't reported to be common, either in these Brazilian gold rush towns or their 19th century American equivalents.

Mining communities, whether in the Amazon or California, weren't seen as a stable social environments-- they're places men go to, to achieve something and then move on

A prevailing social script for the men trying to make their fortune in mining or other frontier activities was not the thought of taking a wife to such a place, rather it was to endure it until they could make enough money to create the ordered domestic environment of the home with more refined entertainments, something different to the drinking establishments and mining camps where they might have spent earlier days

The vast majority of Anglo American gold miners came to California with little intention to settle. Instead, they viewed the Gold Rush as a means of securing the wealth that would allow them to return to the East and establish themselves as independent farmers or businessmen surrounded by a supportive and dependent family . . .

These same qualities were also supposed to typify their leisure pursuits. The rough culture of the laboring class, centered on the tavern, card table, and physicality, was explicitly juxtaposed against “proper” forms of leisure, such as lectures, visits, and balls

Sex ratios and morés changed as the economy developed, so towns did become more "civilized" and less rough and tumble over time, but it would have been an unusual women who favored drinking establishments that were nearly all men-- but the opportunities of the West did attract unusual women, including a few women who dressed and lived as men.

A look at data from Owyhee County in Idaho during the latter half of the 19th century will give you a reasonable idea of the sex ratios that were common in mining boom towns; %20 female was common, and not a level likely to make "casual hanging out" in bars a great idea.

There were many more women on farming communities -- but then, there were many fewer drinking establishments. So if you are thinking of, say, a farm household in Oregon circa 1870 -- the sex ratio might be close to 50-50-- but there'd likely be no "saloon" to "casually hang out in". By contrast if you're thinking of mining town, the sex ratios would be skewed towards men, and you'd have bars, but not many women around.

So I suppose if you're looking to find a women in a saloon socially, you'd be looking for a farming community that didn't frown on saloons. Wives -- and especially widows-- had all sorts of legitimate occupations in the old West, running boarding houses, laundries, hotels, and teaching, among others. Some of them no doubt wanted a drink, and if you look hard enough you'd likely find communities were public drinking by women was more acceptable; but we don't have a lot of reports of this. Far more common are reports of female saloon _owners_ or proprietors, viz:

Further evidence of female saloon ownership in Braidwood can be found in a Wilmington Advocate article from 1886 which tells the tragic story of the destruction by fire of the saloon of Mrs. Carlo Bonino. The 1880 census lists two female saloon owners, Ann Lewis and Margaret O’Donnell . . . Lewis was the fifty-six-year-old wife of a miner, who probably ran the saloon with the help of her family in order to supplement their income. There is plenty of other evidence that women labored in Braidwood’s saloons as saloonkeepers. In 1880 Catherine Garrity and Sarah Burt were arrested for “selling alcohol to a minor,” while Margaret Powell was arrested for “selling alcohol without a license.”

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Sources:

Roberts, J. Timmons, and F. Nii-Amoo Dodoo. “Population Growth, Sex Ratios, and Women's Work on the Contemporary Amazon Frontier.” Yearbook. Conference of Latin Americanist Geographers, vol. 21, 1995, pp. 91–105.

Blackburn, George M., and Sherman L. Ricards. “Unequal Opportunity on a Mining Frontier: The Role of Gender, Race, and Birthplace.” Pacific Historical Review, vol. 62, no. 1, 1993, pp. 19–38.

Boag, Peter. "Re-Dressing America’s Frontier Past", University of California Press:2011

Walsh, Margaret. “Women's Place on the American Frontier.” Journal of American Studies, vol. 29, no. 2, 1995, pp. 241–255.

Barleen, Steven. “The Saloon on the Prairie: The Family and the Saloon in Braidwood, Illinois, 1865-1883.” Journal of the Illinois State Historical Society (1998-), vol. 106, no. 2, 2013, pp. 193–223.

Herbert, Christopher. “‘Life's Prizes Are by Labor Got’: Risk, Reward, and White Manliness in the California Gold Rush.” Pacific Historical Review, vol. 80, no. 3, 2011, pp. 339–368.