This article presents a numerical model for solving the nonlinear random vibrations of offshore moored floating structures under seismic excitation. The offshore moored floating structure consists of the floating platform and mooring cables. The floating platform is considered as a rigid body with 3 degrees of freedom. The nonlinear equations of motions of the mooring cables are established using the nonlinear cable elements that are formulated based on the extended Hamilton principle. The nonlinear hydrodynamic drag forces that act on both the floating platform and cables are considered. In order to carry out the random vibrational analysis, the connection conditions between the floating structure and mooring cables are given to formulate the equations of motions of the whole system. Finally, the moored floating structure under horizontal seismic ground accelerations with Kanai-Tajimi model are analyzed using Monte Carlo simulation method. The probability density functions of the displacements of the moored floating structure and the maximum tensile force in cables are presented. The influences of different sag-to-span ratios or inclined angles of the mooring cables on the mean value and standard deviation of the displacements of the floating structure and the maximum tensile force in cables are analyzed.