If $A$ is a matrix, then $e^{At} \leq C e^{-\lambda t}$ if and only if the spectrum of $A$ consists of eigenvalues with negative real parts.
Is there a similar result, relating stability to the spectrum, when $A$ is a non-selfadjoint operator, when $e^{-At}$ is interpreted as the solution to $$\frac{\partial u}{\partial t} = - A u,$$ with appropriate boundary conditions? I assume that there is, but I haven't been able to find a reference.
You need some assumptions on $A$ to have a unique solution of the evolution equation (and one should also talk about the type of solution). Anyway, I think the answer to your question lies in the Hille-Yosida theorem:
If $A$ is the generator of the strongly continuous semigroup $T$, then $\|T(t)\|\leq M e^{\omega t}$ if and only if the spectrum of $A$ is contained in the half-plane $\{z\in\mathbb{C}\mid \operatorname{Re}z\leq \omega\}$.
In this case $u(t)=T(t)x$ is the unique mild solution of the initial value problem \begin{align*} \dot u(t)&=Au(t),\\ u(0)&=x. \end{align*} The Hille-Yosida theorem also gives a characterization of the operator that generate strongly continuous semigroups.