Having visited there last year, I would certainly recommend lodging in St. Andrews. Another option is to find lodging in the surrounding area, i.e. Crail, Cupar, Leuchars, Dundee, etc, but it will add a bit of inconvenience as far as travelling goes. We stayed at a beautiful self-accommodating lodge in Auchtermuchty, which is a 20 minute drive from St. Andrews. But, we had a car rental and it wasn't Open week, so it was fairly easy to get around.
The major downside to staying in St. Andrews is that your wife is liable to get bored after a day or so if she's not into the golf experience. Mainly because there's just not much there for the non- golfer after you've spent a day seeing the sights and going through the touristy shops. If that's a real possibility, I'd recommend staying in Edinburgh where there's plenty to keep her occupied (and happy!). Also, assuming you haven't been there before, make sure she understands that "shopping" in Scotland is NOT what most women in the States consider "shopping", i.e. no megamalls and strip shopping centers.
Either way, one or both of you should be prepared to spend alot of time commuting. It takes twice as long to get from point A to point B compared to what you're used to. Regardless of the inconveniences you might encounter, it will still be the greatest trip of your life. The people are wonderful, the scenery is breathtaking, and the St. Andrews experience is incredible. I can't wait to get back. Whatever you decide, it will be worth the effort. Have a great time!