If you allow renormings of the space when computing $\beta$ then $\beta(S_X)=2$ implies that $X$ is non-reflexive. This follows (I think) from a [theorem of Odell and Schlumprecht.][1]


  [1]: http://www.ams.org/journals/jams/1998-11-01/S0894-0347-98-00251-3/S0894-0347-98-00251-3.pdf