It's difficult to generalize about the entire working class, but broadly, yes, there was some concern for fashion.

Complaints about domestic servants dressing "above their station" so that people could theoretically mistake the maid for the mistress goes back centuries and certainly existed during the mid-19th century. The American Family Encyclopedia of Useful Knowledge (1856) specifically says of maidservants' clothing:

Of late years, the low price of most of the articles of clothing has introduced into many classes a more showy style of dress than they formerly indulged in, and among maid servants this is particularly evident.

It then goes on to recommend employers not allow their maids to wear silk gowns and stockings, fine laces, feathers, flowers, and jewelry, or to allow them to waste time styling their hair into ringlets, and also gives suggested spending limits to set for maids' clothing consumption. Many other domestic manuals bear out the same viewpoint: maids want to dress too well, and we need to keep them from doing it for practical and moral reasons. (Why Do the Servants of the Nineteenth Century Dress as They Do? is a British publication, but it may be of some interest to you.)

Now, this is only a narrow range of working-class women. They're the easiest to find sources on, because servants' overdressing was of great importance to the middle classes who were writing and publishing at the time, where the dress of factory workers and shopgirls was much less important. When it comes to those who lived in the country, I can't find American sources - British ones, though, continue to talk about "cheap finery", village girls' imitating the local gentry in dress, and sometimes factory girls dressing fashionably outside of work hours. (Women who worked in dressmakers' or milliners' shops were generally expected to dress well in order to do credit to the establishment as houses of fashion.)

As for how they did it, to a certain extent they did it for themselves. Sewing skills were much more widespread than they are today, and women were able to retrim bonnets and alter dresses to fit with current fashions regularly, if they wanted to. There was also a secondhand market (as well as clothing simply being handed down to lady's maids as a perk of the job), and the Industrial Revolution was helping to lower prices on many textile and dress items.