We present low-resolution Keck I/LRIS spectra spanning from 3200-9000 A of nine young brown dwarfs and three low-mass stars in the TW Hya Association and in Upper Sco. The optical spectral types of the brown dwarfs range from M5.5-M8.75, though two have near-IR spectral types of early L-dwarfs. We report new accretion rates derived from excess Balmer continuum emission for the low-mass stars TW Hya and Hen 3-600A and the brown dwarfs 2MASS J12073347-3932540, UScoCTIO 128, SSSPM J1102-3431, UScoJ160606.29-233513.3, DENIS-P J160603.9-205644, and Oph J162225-240515B, and upper limits on accretion for the low-mass star Hen 3-600B and the brown dwarfs UScoCTIO 112, Oph J162225-240515A, and USco J160723.82-221102.0. For the six brown dwarfs in our sample that are faintest at short wavelengths, the accretion luminosity or upper limit is measurable only when the image is binned over large wavelength intervals. This method extends our sensivity to accretion rate down to ~1e-13 solar masses/year for brown dwarfs. Since the ability to measure an accretion rate from excess Balmer continuum emission depends on the contrast between excess continuum emission and the underlying photosphere, for objects with earlier spectral types the upper limit on accretion rate is much higher. Absolute uncertainties in our accretion rate measurements of ~3-5 include uncertainty in accretion models, brown dwarf masses, and distance. The accretion rate of 2e-12 solar masses/year onto 2MASS J12073347-3932540 is within 15% of two previous measurements, despite large changes in the H-alpha flux. ; Comment: Accepted by ApJ. 23 pages, 5 figures, 3 tables