From 1900 until roughly the middle of the century, fashion remained largely the same. Men wore a fedora and suits. Women wore dresses, and occasionally some form of hat.
While widely held, this view is somewhat ignorant. Though the fashions look very similar to us looking back today, there was a great deal of change and evolution of fashion between 1900 and 1950. This is probably more obvious in women (dresses went from hourglass at the turn of the century to the "flat flapper" and back by the 40s, and underwear evolved from a light corset to the girdle, which used elasticized material to achieve a gentler effect. The introduction of the shirtwaist was a Big Deal, becoming essentially the uniform of the working class woman, and the first skirtsuits appear in this period as well, which together with the fact that women were wearing pants for the first time ever (albeit "feminine" pants, very different from men's) were probably the first steps along the path to more similar clothes between men and women. Skirts and pants generally get shorter. We also see hats get massive and then smaller and more practical, and the eventual adoption of something very much like men's hats for women's wear.
Men, for their part, started the century with upper class men wearing something very similar to what we'd called a tuxedo at almost all times, with tailcoats when appropriate and vests almost always. Vests started very high and got lower and lower, eventually starting to disappear entirely by the end of half-century and something closely resembling the modern suit became standard. Working class men, of course, wore less opulent clothing, but still would have probably worn something very like a suit at the beginning of the century, and would likely have progressed to shirt sleeves much of the time by 1950. Top hats, homburgs, bowlers, and "news boys" gave way to fedoras and pork-pie hats. Collars became incredibly high and stiff (often removable), before becoming something more like what we'd see today.
To put it another way: if you told a young lady or young gentleman in the 20s, 30s, or 40s that they dressed like their parents had, they'd probably be even more offended than a modern teen.
Anyway, as Apollo 20 says, it's hard to do, because much of it is driven by technology, funnily enough. T-shirts became possible with the ability to make cloth elastic enough to be drawn over the head like that. The incredibly reduction in the cost of rubber allowed a complete revolution in undergarment-town. It's not exactly part of your prompt, but a lot of the fashion change in the early 20th century was to make things easier to mass-produce.
There's also the effects of the war, of course. A lot of people argue that WWII killed the vest in daily wear, and that the various women's auxiliary services were largely responsible for the influx of men's fashion to women's. Even without its being made into a sex symbol by 50s badboys, wearing a t-shirt and nothing else was already somewhat common place in the mid 50s due to GIs dressing like that in the War.
And then, of course, the counter-culture and hippies completely blasted it open (though interestingly the Mods of Britain actually dressed even more sharply than before on their souped-up scooters).
To prevent the changes you probably meant, a good first step would be to keep the film industry in the Northeast (so that the fashions of sunny SoCal don't get distributed through film). Then keep America out of the War (either, but especially WWII). That should also kill off counter-culture, at least in the short term. Maybe lighter fabrics can be incorporated into more traditional clothes, though then you hit problems like "jersey doesn't hold a crease well".