The invention of the safety bicycle in the 1880s - essentially the bicycle as we know it today, with a diamond frame and chain drive to the rearmost of two equal sized wheels - was an event of some considerable sociological* significance, providing (after a brief period where it was a plaything of high society) the great mass of the population with a form of independent transport that was affordable and did not call for the gymnastic feats and exposure to danger required by its high-wheeled precursors, and unlike them did not require very long legs for optimal mechanical advantage.
Among the groups to benefit from this new mobility were women. The contemporary development of rational dress meant that women were now able to use the new machines and gain an unprecedented facility for independent travel.
This, of course, would never do. The prospect of women in trousers gallivanting around the countryside frightening the horses could not be countenanced by any right-thinking person; inns and hotels refused admission to women in rational dress and the courts upheld their right to do so.
Capitalism works wonders when faced with the possible loss of a market comprising half the population, however; a technical fix was clearly in order to allow ladies to bicycle whilst demurely dressed in ankle-length skirts. The solution was to replace the top tube of the bicycle frame with a tube running more or less parallel to the down tube; the resulting machine could be easily mounted and ridden in a long skirt.
There was, of course, no free lunch. The resulting design is heavier (the sloping tube is longer) and, more problematically, removes structurally important triangulation resulting in a lack of torsional rigidity: the frame tends to twist longitudinally around the bottom bracket. This makes the machine far harder to control as the wheels go in and out of track, particularly at any sort of speed, when they can suffer from speed wobble.
A century later, women's clothing has changed beyond all recognition, but the industry still persists in selling open-frame machines as "ladies/girls/women's bikes". Supposed improvements like the "mixte" frame (with a split tube running from the top of the head tube to the rear hub instead of the sloping top tube) were merely cosmetic; ironically, the actual morphological differences between men and women (particularly women's relatively longer legs and shorter torsos and a general need for smaller machines) were rarely addressed: components were designed for the commonest men's sizes; it is geometrically impossible to use standard road bike components to make a bike small enough for the shortest 25% or so of the female population without some extremely undesirable features, particularly a very high bottom bracket (increasing the actual height of the rider above the ground) and pedals that overlap the front wheel. The development of the mountain bike in a more egalitarian place and time has made for some improvements here, however.
The idea that a women's bike has to have an open frame is sufficiently ingrained that women's organisations in Oxford at one stage complained that the road signs and marking for the city's then new cycle route network only showed "men's" bikes and insisted that half should be "women's" bikes - of the sort that men imposed on them a century earlier.
The open frame does still have its place, however; it enables some people with limited mobility to have a bike that they can get onto, and indeed the use of composite materials to build a monocoque frame can at a price produce a step-through machine which does not suffer from the rigidity problems of a traditional open frame - indeed, prior to the introduction of restrictive regulations by the UCI, many top-end specialist racing machines were produced along those lines.
There are a few (but an increasing number) manufacturers who produce top-end real women's road bikes, without open frames but carefully adapted to suit women's actual physical requirements; pride of place here should probably go to Georgina Terry.
* And indeed biological. People's new ability to travel fairly considerable distances during relatively brief periods of leisure meant that the gene pools of communities that had been all but completely isolated were suddenly opened up to each other, with generally beneficial results. Just one of the reasons why we're all so much better looking than our forebears. But I digress.