The relationship you just described sounds a lot like the one I had when I first met my husband. I can't presume to know where you and she are headed, but I will share my story with you straight if it can be of any help:
When I met my husband we had a bit of a long-distance relationship and I was definitely a "drink at home alone because I was bored" wine drinker. I'm ashamed to admit that since a lot of our courtship was over the phone in the evenings, I don't remember many of those all-important getting to know you conversations. One of my great regrets is that I did not enjoy our relationship fully present and sober.
Had he asked me about it (which he didn't) I would have minimized it completely and hid it better. 100%. But I also hadn't been to rehab and wasn't as forthcoming as your girlfriend sounds. Firestorm is right though - to keep our addiction active we will lie and sweet-talk in ways we'd never dream of under any circumstances. Talking to her is a good first step, but only time and actions will tell if she was being truthful unfortunately.
I will be honest and tell you that as the relationship progressed I got worse - LOADS worse. In the beginning I hid it. I'd have a glass of wine in the open and a bottle of vodka under the bathroom sink. I'm sure he never understood how I could get so drunk off a glass or two. But by the end I was openly binging and causing him so much pain and sorrow I can't ever forgive myself. Like your girlfriend though, it became less and less frequent - down to once every month or two - so the rest of the time I was a great wife and partner. I think that's what makes it really tough for both people to admit what a problem it really is.
That's where I was in my drinking career though. Seven years ago it was a problem and I wasn't ready. All through our marriage it has been there and I have been getting more and more ready. I hope with all my being that this is it. Who knows? She may be ready now. Or not. Only time will tell and you must decide if you want to stick around to find out.
Although I can understand people telling you to RUN! I'm sure glad nobody told my husband to!
But he had to wait many years before I was ready to give it up. Only you and she can determine how it will play out for you.
If you think Alanon or other counselling would be of benefit I think you should go for it. Nobody can predict if she will stop or not, and the odds aren't in your favour given the stats on alcoholism - hence all the "Run!" advice I would imagine. I've been so blessed to have a patient man who loves me and saw what I could be without the alcohol. But he hasn't had the life he deserves for a long time, and there's a lot of damage to be repaired. I wouldn't wish that hurt on anyone, and it kills me to know I caused it for him.
I hope she is ready. And if she is not, there is nothing you can do to make her be ready. Trust me - he tried so I know.
Good luck to you whatever you decide.
Jayne