In this paper, we present the use of the orthogonal spline collocation method for the semi-discretization scheme of the one-dimensional coupled nonlinear Schrödinger equations. This method uses the Hermite basis functions, by which physical quantities are approximated with their values and derivatives associated with Gaussian points. The convergence rate with order and the stability of the scheme are proved. Conservation properties are shown in both theory and practice. Extensive numerical experiments are presented to validate the numerical study under consideration.