We report measurements of branching fractions and charge asymmetries of exclusive decays of neutral and charged B mesons into two-body final states containing a charmonium state and a light strange meson. The charmonium mesons considered are J/psi, psi(2S) and chi(c1), and the light meson is either K or K(*). We use a sample of about 124x10(6) BB pairs collected with the BABAR detector at the PEP-II storage ring at the Stanford Linear Accelerator Center.