Due to the atomic level thickness and novel properties, two-dimensional (2D) materials have received extensive attention on the research and application of future nanodevices. In this paper, the mechanical and electronic properties of $\ensuremath{\alpha}\text{\ensuremath{-}}{M}_{2}{X}_{3}$ ($M$ = Ga, In; $X$ = S, Se) monolayers are studied to explore their applications in 2D electronic devices. First-principles calculations based on density functional theory indicate that these four $\ensuremath{\alpha}\text{\ensuremath{-}}{M}_{2}{X}_{3}$ monolayers are all semiconductors and possess Young's modulus of less than 100 ${\mathrm{N}\phantom{\rule{0.16em}{0ex}}\mathrm{m}}^{\ensuremath{-}1}$ with a deformation range up to about 30%. In addition, the carrier mobility of the $\ensuremath{\alpha}\text{\ensuremath{-}}{M}_{2}{X}_{3}$ monolayer exceeds 600 ${\mathrm{cm}}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{V}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ and remains high under strain. In particular, because the band edge shifts under compressive strain, the electron mobility of the $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{S}}_{3}$ monolayer increases to about 1800 ${\mathrm{cm}}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{V}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ at $\ensuremath{-}3%$ strain, which is approximately three times the value without strain. The excellent ductility and strain-promoted electronic properties make the 2D $\ensuremath{\alpha}\text{\ensuremath{-}}{M}_{2}{X}_{3}$ promising candidates for the application of flexible electronic devices.