Quasi-elastic electron scattering on the deuteron is a benchmark reaction to test our understanding of deuteron structure and the properties and interactions of the two nucleons bound in the deuteron. The experimental data presented here can be used to test state-of-the-art models of the deuteron and the two-nucleon interaction in the final state after two-body breakup of the deuteron. Focusing on polarization degrees of freedom, we gain information on spin-momentum correlations in the deuteron ground state (due to the D-state admixture) and on the limits of the Impulse Approximation (IA) picture as it applies to measurements of spin-dependent observables like spin structure functions for bound nucleons. We measured the beam-target double spin asymmetry for quasi-elastic electron scattering off the deuteron at several beam energies using the CEBAF Large Acceptance Spectrometer (CLAS) at the Thomas Jefferson National Accelerator Facility. The deuterons were polarized along (or opposite to) the beam direction. The double spin asymmetries were measured as a function of photon virtuality , missing momentum, and the angle between the (inferred) "spectator" neutron and the momentum transfer direction. The results are compared with a recent model that includes Final State Interactions (FSI) using a complete parameterization of nucleon-nucleon scattering, as well as a simplified model using the Plane Wave Impulse Approximation (PWIA). We find overall good agreement with both the PWIA and FSI expectations at low to medium missing momenta, including the change of the asymmetry due to the contribution of the deuteron D-state at higher momenta. At the highest missing momenta, our data clearly agree better with the calculations including FSI.