I think it's fine to ask the price for a small number of drinks, but not the entire menu one by one. I undersand why many places wouldn't have a menu listing all possible drinks and don't really have a problem with that. For a printed menu of specially drinks, it does seem kind of ridiculous not to show prices. I can't quite see asking, what's your least expensive drink.
In general, the price variation for house cocktails (alcohol and a mixer), say gin and tonic, vodka tonic, rum and coke, vodka cranberry, is within a dollar or two, so it's unlikely to get you much savings by asking about each one, so it might be easier to ask just about one that you are interested in. The next price point might be martinis since in many places they have more alcohol. Another category is specialty drinks, the ones listed on a menu. Finally, you get into premium brands, for those, it's worth asking, particularly if you are talking about a very special scotch or Cognac which can get very expensive and vary quite a lot.
For beer, there are probably three or four categories, inexpensive domestic vs. premium (microbrews, imported, etc) on tap and then the same two categories in a bottle. Within each category, there is not usually much variation, say a dollar or two, so asking for the price of each beer is a bit much, I think.
Wine is a whole different issue, with the potential for huge variation and a wine menu with prices is key for all but the house wines, where you could certainly ask.