Images have been obtained at the 6-m telescope at the Special Astrophysical Observatory (SAO) of the Russian Academy of Sciences in the Hα line and in the continuum for 20 dwarf spheroidal satellites of M31: And XI-And XXX, plus the distant Globular cluster Bol 520. Their star formation rates (SFR) are estimated using the Hα flux and the ultraviolet FUV flux measured with the GALEX space telescope. Most of the dSph satellites of M31 have extremely low star formation rates with a characteristic upper limit of SFR ~ 5 × 10-7. We have made similar estimates of SFR from the Hα and FUV fluxes for 13 galaxies with low surface brightness recently discovered in the neighborhood of M81. Eleven of them are physical satellites of M81 with typical SFR < 1 × 10-5. The median stellar masses of these satellites of M31 and M81 are 0.9 and 1.9 million solar masses, respectively. Our Hα observations place a 2-3 times stricter limit on the value of SFR than the data from the GALEX satellite, with a substantially higher (3-5 times) angular resolution.