We investigate the effect of Co^{2+} (spin-1/2) impurities on the magnetic ground state and low-lying spin excitations of the quasione-dimensional spin-1/2 antiferromagnet SrCuO_{2} by means of neutron scattering, muon spin spectroscopy, and bulk (ac and dc) magnetic susceptibilities. We found that dilute Co doping induces an Ising-like anisotropy and enhances the magnetic ordering temperature rather significantly, but preserves the gapless nature of the spin excitations. These results are in apparent contradiction with the recent studies of Ni (spin-1) doped SrCuO_{2}. Low-temperature magnetic behavior of the Co-doped zigzag chains in SrCuO_{2} reveals the presence of a weak geometrical spin frustration.