First of all most golfers don't keep a handicap. But here is the distribution of handicaps (USGA). Looking at these data one would have to guess that average is somewhere near 13. However since I've seen results that say most golfers can't break 100 playing by the rules I somehow suspect people who keep handicaps are the better players and don't represent golfers at large. There are likely some marketing organizations that keep track of scores of as many golfer as they can via surveys but I'd bet you can't get these data for free as they spend money getting these data.
guess I should have added the site. http://www.usga.org/handicapping/articles_resources/Men-s--Handicap-Indexes/