How can I deal with an alcoholic boyfriend who suffers from social anxiety? How can I help you?
Ask yourself first why you want to "deal with" an alcoholic boyfriend. Look at how you worded your question. If we love someone and it's a healthy relationship, we don't normally say that we are "dealing with" them. We want to be around them and it feels like it's adding to our lives rather than taking away from it.
To answer your question, I would say that you can recommend that he seeks professional counseling or joins an anonymous program like Alcoholics Anonymous where he can be around others that face the same struggles. If you want to stay in the relationship, be supportive while still valuing yourself and making sure your own needs are being met. It is easy to get caught up in supporting someone else that we care about as they are facing their own challenges and we forget to take care of ourselves. If you drink alcohol, consider quitting yourself and/or remove temptation from the home if you live together.
It is hard to work through alcoholism but it is possible. I have been sober for almost 3 years now. But, remember that he has to want to quit drinking, you can't force him to.