We investigate the stability of the first excited state, the so-called "pi-state," of Bose-Einstein condensates in a double-well potential. From the condition of complex excitation energies, we determine the critical barrier height, above which the pi-state is dynamically unstable. We find that the critical barrier height decreases monotonically as the number of condensate atoms increases. We also simulate the dynamics of the pi-state by solving the time-dependent Gross-Pitaevskii equation. Our simulation results show that the pi-state in the dynamically unstable region exhibits distinctively different behavior from that in the dynamically stable region.