Get her on a full suspension bike. Yeah, you probably don't want to buy one for her right away since she may not be into it. So, rent or demo a good one from the bike shop and make sure it fits her great. Then make sure the suspension is set up just right for her. Finally, tire pressure is very critical, it makes a huge difference in ride and traction and getting it right will increase her confidence and give you a greater chance of success. My wife rides her tires at about 25 psi. Rent or demo a few times and take her on some easier, smooth trails with good scenery. If she decides she likes it and you get her her own FS bike then it is your responsibility (and in your best interest) to ALWAYS make sure her equipment is in tip top shape and working perfectly. I do whatever I can to make my wife's bike as comfortable and enjoyable as I can. If upgrading something will help her perform better, be more comfortable or have more fun, consider it done. Get her a nice ride, don't skimp on her gear!
This is how my wife got passionate about mtn biking. She had a full rigid bike for a few years and went on a few mountain rides, but never really got into it until we got her on a nice FS demo bike a few times. Then we decided that it was worth it to shell out the $$ for her own FS mtn bike. She's now REALLY been mtn biking for 8-9 years and LOVES it, and has really become an awesome rider! She can't get out often enough. In the two races she did this year she won both by very large margins!
When you do get her out on a ride, don't be barking at her what to do in every situation (stand up, get over those bars, get your butt behind the seat, etc.). With my wife it worked better if we stopped before a challenging section and talked about 1 or 2 skills needed to make it through (and why they are needed) so they were fresh in her head and she could try them when she felt comfortable doing so. It also may be helpful to let her try it on her own and if she fails then you can say, "it helps me get up that steep section when I lean over the bars and scoot way up onto the nose of the saddle because you need to get your weight forward to keep the front of the bike down, but you still need some weight on the rear tire for traction." Sometimes it works best if I wait for her to ask what she should be doing differently.
She hated it when I rode behind her (she felt too much pressure). She much preferred to follow me (it will help her see how you pick your line). Go slow, stop often, and talk.
As a side note, over the last couple of years she has been introducing some of her lady friends to mountain biking and has found that some of them do better in an all-girls setting. I think it all has to do with patience and understanding (some of their husbands have neither).
Anyway, these are just some suggestions that have worked successfully for us. I hope they work out for you, because I'd rather ride with my wife than anyone else!