It sorta depends what end of the common you're at, but BWX isn't much of a walk from the downtown crossing end (i.e., close to jm Curley and Stoddards) and Charles Street Liquors isn't too far from the Beacon Hill end. The Tip Tap Room and Kinsale are also fairly close
There's this restaurant called Rattlesnake at Poe's Kitchen that I think is on Boylston and not far from the common. The food was crap but they tap list was great and the bartenders were great to talk craft with and would happily give out samples, etc. Been a while since I've been in Boston though.
Boston Wine Exchange on Devonshire St. and Charles St. Liquors are probably the best stores in the area for craft beer selections. Stoddards and Curley's are good options too. I'll also suggest the Parish Cafe on Boylston St. It's across from the Rattlesnake and has some descent beers and fantastic sandwiches designed by top chefs.
Curley's has been around a couple years. One review here: http://www.beeradvocate.com/beer/profile/29141/?view=ratings
I echo the recommendation for Stoddards, JM Curley, and Poe's Tip Tap Room. While not exactly in the Boston Common area, Bukowski's is a few stops down on the Green line (or maybe a 15-20 minute walk) and has an excellent local draft selection as well.
Parish Café on Boylston, decent beer list, and great sandwiches. Baco on St. James usually has some interesting bottles.
Boston is tiny, don't limit yourself to one specific neighborhood. JM Curley is solid but a few minutes in any direction will get you somewhere at least as good, and as far as shops go, BWX will always have that "location mark-up". Salty Pig and/or Coda. Both very close to Urban Grape. EDIT: Merril and Co also has a nice list... Row 34 and Pastoral. Both very close to Bees Knees, Trillium and another solid shop (can't remember the name). People tend to underestimate how close everything is to everything here.
I forgot to mention Picco in the South End. They have awesome pizza and ice cream, plus an impressive draft list. Depending on where you are, they're about a 10 minute walk from the Common.
