Unstable or optimally growing perturbations of turbulent flows are often representative of the energy-containing coherent structures populating the flow, as for streaks in a turbulent channel. Within this framework, this work aims at studying the modal and nonmodal stability of stably stratified turbulent channel flow, assessing the influence of stratification while changing the friction Richardson number, Riτ, at fixed friction Reynolds number, Reτ. When increasing the stratification of the flow, the energy gain for streamwise independent perturbations at the outer peak increases by two orders of magnitude, and the spanwise wavenumber for which the energy gain peaks reaches values comparable to those reported in the direct numerical simulations of Garcia-Villalba and Del Alamo. At the same time, the value of the optimal gain for the inner peak slightly changes, corroborating the observations made through direct numerical simulation (DNS) about the fact that the wall cycle is not altered by the presence of stratification. Moreover, for nonzero values of the streamwise and spanwise wavenumbers, α and β, the energy gain curve has two peaks, one for shorter target times and α > β, leading to a center-channel temperature peak, and another occurring for α < β at larger target times. In the former case, energy production is mostly linked to velocity production, whereas, in the latter case, the strongest term is that of temperature production, indicating that this mechanism is driven by the increase of the potential energy rather than the kinetic one, and it is intimately linked to the presence of stratification. For strong stratification, the optimal energy gain considerably extends towards higher values of α, leading to energy amplifications reaching three orders of magnitudes for values of α up to 2. The associated optimal perturbations are characterized by temperature patches at the center channel, phase lagged by π/2 with the wall-normal velocity, similarly to gravity waves recovered in the DNS for sufficiently large stratification. However, for large values of β, we observe an increasing asymmetry in the optimal perturbations, probably due to the shielding effect of the core of the channel, as also observed in the DNS of Garcia-Villalba and Del Alamo.