Basis set methods for calculating dynamic polarizabilities and excitation energies via coupled Kohn-Sham equations within time-dependent density functional theory are introduced. The methods can be employed after solving the ground state Kohn-Sham equations with a fitting function approach. Successful applications of the methods to test molecules are presented. Coupled Kohn-Sham methods based on the linear response of the Kohn-Sham density matrix are derived from the standard coupled Kohn-Sham equation based on the linear response of the electron density and the relations between the two types of coupled Kohn-Sham equations are investigated. The choice of norm functions associated with basis set representations of the coupled Kohn-Sham equations is discussed and shown to be a critical point of basis set approaches to time-dependent density functional theory.