  We study on which compact Sasakian 3-manifolds the Reeb field, which is a
Beltrami field with eigenvalue 2, is an energy minimizer in its adjoint orbit
under the action of volume preserving diffeomorphisms. This minimization
property for Beltrami fields is relevant because of its connections with the
phenomenon of magnetic relaxation and the hydrodynamic stability of steady
Euler flows. We characterize the Sasakian manifolds where the Reeb field is a
minimizer in terms of the first positive eigenvalue of the curl operator and
show that for $a>a_0$ (a constant that depends on the Sasakian structure) the
Reeb field of the $\mathcal{D}$-homothetic deformation of the manifold with
constant $a$ (which is still Sasakian) is an unstable critical point of the
energy, and hence not even a local minimizer. We also provide some examples of
Sasakian manifolds where the Reeb field is a minimizer, highlighting the case
of the weighted 3-spheres, on which another minimization problem (for the
quartic Skyrme-Faddeev energy) is shown to admit exact solutions.
