please expand by clicking the graphic.
Notes after the jump.
To answer the question posed in the title, some clarification is in order. "How common is a one-WAR player?" Well, if you're saying "exactly one" or "1.0," not very common. In fact, there are only TWELVE exactly-one-WAR players. It's my belief, though, that the question isn't really asking that and is instead attempting to find the commonality of a range of slightly-above average players.
To do just that, I grouped the data into sections which include every tenth's place after zero for the integer in question. There were 18 players between -2 and -1, 326 between -1 and 0, and so on. In case it's not explicit, each circle represents a single player at a single one-tenth iteration of a particular WAR. For example, the section at the top of the graphic which is listed }5 represents all players from 7.0 - 7.9* Wins Above Replacement (of which there are 5). Where there is a grouping of players on a single line (see the zero line), each has the same WAR.
Back to the question in the title. The better answer is, "38% (369) of all players in 2010 earned between zero and one WAR, while only 83 players (9%!) earned between one and two." Clearly and obviously, and I think the entire scope of the curve captures this well, players that earn greater than one WAR are increasingly rare.
Data sourced from Fangraphs, "all players" for 2010.
* Josh Hamilton's 8.0 was accidentally included in the 7.0-7.9 grouping. All other integer groupings start at x.0 and go to x.9.