In this paper, we investigated the Hyers–Ulam stability of the coefficient multipliers on the Hardy space $H^2$ and the Dirichlet space ${\mathcal{D}}^2$. We also investigated the Hyers–Ulam stability of the coefficient multipliers between Dirichlet and Hardy spaces. We provided the necessary and sufficient conditions for the coefficient multipliers to have Hyers–Ulam stability on Hardy space $H^2$, on Dirichlet space ${\mathcal{D}}^2$, and between Dirichlet and Hardy spaces. We also showed that the best constant of Hyers–Ulam stability exists under different circumstances. Moreover, some illustrative examples were discussed.