Very recently, carbon-boron clathrate $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ has been successfully synthesized, in which carbon and boron atoms form $s{p}^{3}$-bonded truncated octahedral cages. Interestingly, the $s{p}^{3}$-hybridized $\ensuremath{\sigma}$-bonding bands are partially occupied. This may drive $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ into a superconducting state, like boron-doped diamond. By means of density functional first-principles calculations and Wannier interpolation technique, we have investigated the electron-phonon coupling and phonon-mediated superconductivity in $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$. Our calculations reveal that there exists strong coupling between $s{p}^{3}$-hybridized $\ensuremath{\sigma}$-bonding bands and boron-associated ${E}_{g}$ phonon modes. Based on the Migdal-Eliashberg theory, we self-consistently solve the anisotropic Eliashberg equations. It is found that $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ is a single-gap superconductor, with superconducting transition temperature being 40 K. The anisotropic ratio of superconducting energy gap is computed to be 32.8%. Further replacing Sr with Ba, the transition temperature can be boosted to 43 K in $\mathrm{Ba}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ due to phonon softening. These findings suggest that $\mathrm{Sr}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ and $\mathrm{Ba}{\mathrm{B}}_{3}{\mathrm{C}}_{3}$ are phonon-mediated high-temperature anisotropic $s$-wave superconductors.