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 transition and activate transverse responses in representative altermagnets RuO₂ and MnF₂. For pristine RuO₂ and MnF₂ with Néel vector n∥ [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 RuO₂ and to a fully compensated ferrimagnetic phase in semiconducting MnF₂. 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. These responses exhibit a clear strain dependence and become progressively stronger 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.