While he probably adds a few runs with his stealing, he gives a LOT back when he removes himself on the basepaths.
The run expectation table values a runner at 1st with no outs at .9116 runs. The figure raises slightly to 1.1811 when he successfully steals second. However, the expected runs total plummets to .2783 if he is caught stealing a base. The run expectation is an aggregate of runs that scored, from that base/out state, to the end of that inning. Therefore, the loss of a baserunner in front of Bay, Martinez, and Youkilis, who are all well above average hitters, is substantially worse.
I believe that the stolen base is a useful tactic, albeit an extremely overrated one. It is better using that tactic in front of players who don't have the ability to score runners from 1st. Losing Ellsbury in front of Green does not hurt as badly as losing him in front of Pedroia.