It is argued that the extended mode-coupling theory for glass transition predicts a dynamic crossover in the α-relaxation time and in the self-diffusion constant as a general implication of the structure of its equations of motion. This crossover occurs near the critical temperature T(c) of the idealized version of the theory, and is caused by the change in the dynamics from the one determined by the cage effect to that dominated by hopping processes. When combined with a model for the hopping kernel deduced from the dynamical theory for diffusion-jump processes, the dynamic crossover can be identified as the fragile-to-strong crossover (FSC) in which the α-relaxation time and the self-diffusion constant cross over from a non-Arrhenius to an Arrhenius behavior. Since the present theory does not resort to the existence of the so-called Widom line, to which the FSC in confined water has been attributed, it provides a possible explanation of the FSC observed in a variety of glass-forming systems in which the existence of the Widom line is unlikely. In addition, the present theory predicts that the Stokes-Einstein relation (SER) breaks down in different ways on the fragile and strong sides of the FSC, in agreement with the experimental observation in confined water. It is also demonstrated that the violation of the SER in both the fragile and strong regions can be fitted reasonably well by a single fractional relation with an empirical exponent of 0.85.