We present the results of a high-resolution imaging survey for brown dwarf binaries in two open clusters. The observations were carried out with the Wide Field Planetary Camera 2 on board the Hubble Space Telescope. Our sample consists of eight brown dwarf candidates in α Persei and 25 brown dwarf candidates in the Pleiades. We have resolved four binaries in the Pleiades with separations in the range 0 &farcs; 094–0 &farcs; 058, corresponding to projected separations between 11.7 and 7.2 AU. No binaries were found among the α Per targets. Three of the binaries have proper motions consistent with cluster membership in the Pleiades cluster, and for one of them we report the detection of H α in emission and Li I absorption obtained from Keck II/ESI spectroscopy. One of the binaries does not have a proper motion consistent with Pleiades membership. We estimate that brown dwarf binaries wider than 12 AU are less frequent than 9% in the α Per and Pleiades clusters. This is consistent with an extension to substellar masses of a trend observed among stellar binaries: the maximum semimajor axis of binary systems decreases with decreasing primary mass. We find a binary frequency of two binaries over 13 brown dwarfs with confirmed proper-motion membership in the Pleiades, corresponding to a binary fraction of 15%. These binaries are limited to the separation range 7–12 AU, and their mass ratios are larger than 0.7. The observed properties of Pleiades brown dwarf binaries appear to be similar to their older counterparts in the solar neighborhood. The relatively high binary frequency (≥ 10%), the bias to separations smaller than about 15 AU, and the trend to high mass ratios (q≥ 0.7) are fundamental properties of brown dwarfs. Current theories of brown dwarf formation do not appear to provide a good description of all these properties.