Actually, during the Dark and Early Middle Ages, many women rode astride, just as men. It's easy to find paintings, especially of Italian women, riding cross-saddle (astride) but also of other European women as well. A notable illustration of the Wife of Bath from the Canterbury Tales shows her riding astride. It seems that early sidesaddles were most often used for ceremonial occasions or by members of the nobility/royalty. A good reference for early sidesaddles is Le Costume du Moyen Age Apres Les Sceaux and Lida Bloodgood's classic, The Saddle of Queens. The first of these features early French royal seals, a few showing noblewomen/royalty riding aside and astride. (My spelling may be a little off), while the second gives a general history (unfortunately not always well documented) of sidesaddles.
The true hey-day of the sidesaddle began in the 18th century and continued into the early 20th. During this time, several important Queens and Empresses were noted riders, and made aside horseback riding popular for women. During the latter half of the 19th century, inexpensive sidesaddles were available, making it possible for women of lower social status to afford them. During this time the quality of sidesaddles ranged from very poor to very, very high, and there were several styles and trees (the wooden foundation of a saddle) to choose from.
The safety of aside riding has been much debated by a number of authors. My general impression from a variety of late 19th century and early 20th century sources is that the sidesaddle is a very safe way to ride in terms of the security of the seat....especially when compared to an English saddle. Most 19th/20th century riding instructors claimed that a woman was much less likely to fall when correctly seated on a sidesaddle than when riding astride, and many modern enthusiasts agree.
That being said, the sidesaddle presents some safety issues that differ from astride riding:
1) The rider is perhaps more dependent on the safety of the saddle itself. If the saddle turns or the girth(s) breaks, then the rider is going off the horse.
2) Riders before the early 20th century were often put in danger by their long, flowing skirts. These skirts could catch on the horns of the saddle, causing the rider to be drug if the horse ran away. The modern backless "apron" prevents this. There was also some danger of the extremely long skirts of the mid-19th century being caught by the horse's feet when he cantered.
3) The danger in aside riding is not so much the danger of falling from the saddle as it is in the potential for being trapped in the saddle if the horse falls onto one side. A rearing horse should never be used for aside riding for this reason: the rider cannot quickly and safely dismount, and she may pull the horse over on her.
4) At least one early author claimed that aside riding was more dangerous on mountain paths because any sidesaddle might turn. While the saddle "turning" problem should not be as much of a problem for a properly girthed, modern sidesaddle perhaps even with breast-collar and crupper, mountain riding such as this is not easy riding for anyone.
5)It must be remembered that even into the early 20th century many inexpensive sidesaddles, such as were sold by Sears or Montgomery Wards, did not have the leaping horn or a balance girth or even a safety stirrup. These are usually considered necessary safety features today. Without the leaping horn, the rider could easily be trotted or bounced off the horse. Of course, many of the ladies' horses then were either well-trained, very old, or not as well-fed as our fat and sassy steeds today! Without the balance girth, the back portion of the saddle could turn, and of course the safety stirrup prevents the rider's foot from being caught in the stirrup.
For a discussion of sidesaddle technique (and the changes in technique) there are a number of antique and vintage books, many of which have now been reprinted for the convenience of the modern aside rider. There are also some modern books on aside riding. These include such works as The Habit and the Horse, Riding and Driving for Women, Sidesaddle, For Whom the Goddess, The Fair Lady Aside and many more. Organizations such as the American Sidesaddle Association often sell these works, or have links to where they can be purchased, and there are at least two free ebooks on aside riding available from Project Gutenberg.