Stacked van der Waals magnets provide a tunable route to altermagnetism, a compensated magnetic order characterized by momentum-dependent spin splitting, and to the topological magnetic excitations that such order can host. In bilayer CrI₃, first-principles calculations and linear spin-wave theory reveal stacking-controlled altermagnetic order and associated magnon band topology. Combining band-representation analysis with calculations of the dynamic structure factor relevant to inelastic neutron scattering, we further characterize the chirally split topological magnons and quantify their energy corrections and lifetimes using a many-body Green's-function approach. The interlayer magnetic ground state is highly sensitive to the stacking geometry, thereby controlling the magnon band topology and transport responses. We further show that magnon–magnon interactions renormalize the magnon dispersion and dynamic structure factor, with a particular focus on magnon decay. Using van der Waals bilayer CrI₃ as a representative platform, our results establish stacking engineering as a structural route for tuning altermagnetism and associated topological magnon excitations, opening avenues toward stacking-controlled spintronic and magnonic devices.