We report magnetic susceptibility, resistivity and heat capacity measurements on single crystals of the Sr(Fe1-xCox)2As2 and Sr1-yEuy(Fe0.88Co0.12)2As2 series. The optimal Co concentration for superconductivity in Sr(Fe1-xCox)2As2 is determined to be x ~ 0.12. Based on this we grew members of the Sr1-yEuy(Fe0.88Co0.12)2As2 series so as to examine the effects of well defined, local magnetic moments, on the superconducting state. We show that superconductivity is gradually suppressed by paramagnetic Eu2+ doping and coexists with antiferromagnetic ordering of Eu2+ as long as Tc > TN. For y >= 0.65, TN crosses Tc and the superconducting ground state (as manifested by zero resistivity) abruptly disappears with evidence for competition between superconductivity and local moment antiferromagnetism for y up to 0.72. It is speculated that the suppression of the antiferromagnetic fluctuations of Fe sublattice by coupling to the long range order of Eu2+ sublattice destroys bulk superconductivity when TN > Tc.