If a $k$-Banach algebra homomorphism $\phi$ is finite as homomorphism of $k$-algebra, and $\mathcal{A}_{1}$ is Noetherian, then $\phi$ is automatically Banach finite: there is a surjective $\mathcal{A}_{1}$-module homomorphism $p:\mathcal{A}_{1}^{\oplus n}\to\mathcal{A}_{2}$, by Proposition 2.35 $\ker(p)$ is closed. Then $p$ is continuous hence is admissible by Corollary 2.5. So $\mathcal{A}_{2}$ is a Banach finite $\mathcal{A}_{1}$-module.