The most applied ab initio method to determine the transition energies to electronically excited states, namely time-dependent density functional theory (TD-DFT), is now well mastered, but alternative approaches can be necessary to circumvent some of the specific limitations of TD-DFT. In that framework, the Bethe-Salpeter method, following a GW calculation of quasiparticle energies, is certainly an attractive approach with the same scaling with system size as TD-DFT, though its accuracy remains to be completely defined, at least for molecular systems. In the present work, we investigate the performances of a partially self-consistent BSE/GW approach in which the starting DFT eigenvalues (but not eigenvectors) are updated. We particularly focus on the convergence of the calculations, the residual impact of the starting eigenstates and the selection of an adequate atomic basis set. This study provides some guidelines allowing to apply BSE/GW in a computationally efficient way and also highlights some limitations of the non-self-consistent BSE/G(0)W(0) approach. [GRAPHICS] .