Suprising to most is the fact that riding in a sidesaddle is actually quite secure. Before I rode aside I thought for sure it was going to feel like I was going to fall off the off-side (right) because there was no leg there. Where in actuality I felt a little insecure to the near-side (left) as both legs were to that side. Once I learned how to sit properly and figured out my balance (which didn't take long) I felt quite comfortable up there. There are some key points to remember when riding aside as I've been told and discovered myself.
Your body and seat should still be square to the horse's, the same as you would when riding astride. Think of keeping your right shoulder back. If your stirrup isn't quite the right length (most likely too short), it can make it difficult to sit squarely. Also think of having a tack under your left seatbone so that you have a good majority of your weight on the right seatbone. This balances out the fact that you have the weight of both of your legs to one side. I've had people tell me that some ladies that are very good at sidesaddle riding would sit so hard on that right seat bone that they'd wear out that side of the seat first. Your body shouldn't be tipped forward or leaning back, your weight should be centered over the mid-point of your right thigh actually. I borrowed this picture from the ISSO's website.The left leg pretty much just dangles and sits in the stirrup. It is used as an aid at times but unlike an astride saddle, you don't put much weight in the stirrup at all. This can cause you and your saddle to be unbalanced. You should have enough space between the top of your left thigh and your leaping head to fit your hand in. You don't want it too tight or you won't have any room to move, it may be uncomfortable and you might not be able to get out if you need to. Your left leg should be long with the heel slightly down.
Your right thigh should be in line with the horse's spine. If it's not, it can make you unbalanced and struggle to keep your body square with the horse (as you would riding astride). If you find your leg is sitting too far to the left, you can get what's called a "queen" to fix this. It's basically just some padding that you wrap around the upright head that keeps your leg a bit more to the right. Another issue can be small people and a saddle that has wide pommels. It can be hard for small/short legs to be comfortable around a really wide pommel. The only remedy for this is to try a different saddle. Your right leg pretty much does all of the work to keep you on. To gain "purchase" or grip on your saddle you push your right knee towards the upright head and your foot/thigh is pushed towards the horse's shoulder. To help engage the correct muscles for the most grip, make sure your right toe is pointed down and your heel is as close to your left leg as possible.Your hands should be very quiet and sit either in your lap or to each side of your right knee. Some people with really big horses or shorter arms may need to purchase longer reins so they can sit comfortably and not have to reach.
The "emergency grip" is one of those things that is important to know how to do if your horse is acting up. Part of the reason you don't want your stirrup too long is so you can effectively do this. Basically you engage your right leg to get as much purchase as you can. With your left leg you want to bring your heel up (while maintaining your toe in the stirrup) so that you can push your left thigh up into the leaping head. Sit up tall and remember to keep that right shoulder back.
Some other key things I've learned. Sidesaddles are much more secure on a horse that has a good set of withers on them. Round, flat withered horses tend to cause sidesaddles to slip sideways if your girth isn't really really tight.
I prefer to ride in a saddle that has a doeskin seat and pommels. I rode in a sidesaddle last summer that had a smooth seat and I felt quite insecure in it. I was told I looked like I was rowing a boat because I had trouble keeping centered and kept having to "pull" myself forward into the saddle because I kept sliding all over. It wasn't that it was a bad saddle but I was (am?) a beginner and my horse has a BIIIIG canter!
I'd say it's pretty secure! Look at Mrs. Esther Stace jumping her horse over a record 6'6"! Amazing!Or this lady on her slighly frisky horse!