Collinear antiferromagnets with broken time-reversal symmetry have emerged as a fertile platform for spintronics. Using a general tight-binding model and first-principles calculations, we show that strain engineering provides a simple route to control magnetic phase transitions and activate transverse responses in representative altermagnets RuO${}_2$ and MnF${}_2$. For pristine RuO${}_2$ and MnF${}_2$ with N'eel vector $𝐧\parallel$ [001], symmetry constrains the off-diagonal elements of the Hall conductivity tensor to vanish, thereby forbidding anomalous transport and magneto-optical responses. Shear strain applied along the $ac$ direction preserves the spin symmetry relating the two spin-opposite magnetic sublattices and therefore maintains the altermagnetic phase. By contrast, shear strain applied along the $ab$ direction breaks this spin symmetry and drives a transition from an altermagnetic phase to a partially compensated ferrimagnetic phase in metallic RuO${}_2$ and to a fully compensated ferrimagnetic phase in semiconducting MnF${}_2$. In addition, the lowered symmetry enables finite anomalous Hall, anomalous Nernst, and anomalous thermal Hall conductivities, as well as magneto-optical rotation angles, which are prohibited in the pristine systems. Within the calculated energy ranges, these responses generally exhibit more pronounced spectral features as the strain amplitude increases. Our results establish strain engineering as an effective route to manipulate magnetic phases and functional responses in unconventional antiferromagnets, thereby expanding opportunities for antiferromagnetic spintronics and magneto-optical applications.