Men’s fashions don’t change that much over time, so it should be easiest to clothe the men in the cast. And since most of the male characters in the play are of the same breed—tough talking gamblers—they can be mostly dressed the same way. Suits. Ties. Pressed shirts. Vests. Fedora hats. Tweed and pinstripe suits were popular at the time, and usually, men wore their suits tight-fitting. The iconic zoot suit, though popular during the 1930s and 40s, might be an interesting choice. It was also fashionable to work bright colors into the ensemble.
For the women, a simple flapper theme might be best and easiest to approximate, though it certainly wouldn’t be appropriate for all characters. Sarah Brown and the women from the Mission should be dressed much more modestly. The flapper style, most appropriate for the Hot Box girls, was characterized by loose-fitting dresses that stopped around the knee (or just above), stockings, and short, bobbed haircuts. Fur coats were very chic, and a mink scarf is an appropriate costume element for the number “Take Back Your Mink” in Act 2.
Though many of these costumes can be pieced together from existing costume closets, for those productions with a little more money and no ability to produce clothing, it might not be a bad idea to rent costumes. The end result may look more authentic, and there are numerous costume rental companies online.
Musical Theatre International has a community rental forum on their website that allows current production teams to rent appropriate costume pieces from past production teams.
The most important tip, however, is to look at pictures. Look at pictures of past productions (especially from Broadway) and at pictures from stylish clothing magazines of the period. These photos can all be found online.