The geodesic motion of pseudo-classical spinning particles has been analyzed in the de Sitter and asymptotically de Sitter spacetimes. Obtaining the conserved charges by solving the generalized Killing equations for the configuration space of the spinning particle, we analyzed the motion of a Dirac fermion on a cone and a plane. The results show spin dependences of the time-dilation and of the orbits of the particles in a gravitational field.