A finite basis set of B-spline functions is adopted to investigate atomic hydrogen with exponential-cosine screened Coulomb potentials. Bound-state energies, oscillator strengths, and multipole polarizabilities are calculated and compared with the available theoretical data. Our results, varying with the screening parameters of exponential-cosine screened Coulomb potentials, are presented. The influence of the screening effect on the energies, oscillators, and polarizabilities is discussed.