A total of 25600 3-inch photomultiplier tubes (PMTs), along with their corresponding frontend electronics, have been installed at the Jiangmen Underground Neutrino Observatory (JUNO). These electronics are housed in 200 stainless steel boxes that operate underwater. To verify the sealing integrity of the underwater boxes following integration, we developed an SF$_6$-based leak test system, opting against the typical helium-based system due to helium's ability to penetrate the PMT glass. After a few hours of accumulating leaking SF$_6$ from the underwater boxes, a leak rate detection limit of $2.3\times{10}^{-9}$~Pa$\cdot$m$^3$/s in terms of SF$_6$ was achieved, corresponding to $1.65\times{10}^{-8}$~Pa$\cdot$ m$^3$/s helium equivalent. This meets the sensitivity requirement of 1$\times$10$^{-7}$~Pa$\cdot$m$^3$/s. This system was critical in identifying and replacing a few cases of leaking underwater boxes before installation.