Let $(M,\omega)$ be a compact symplectic manifold, and $J$ be a compatible almost complex structure. Given a $1$-periodic Hamiltonian $H: M \times \mathbb{R} \to \mathbb{R}$, we define the action functional $A_H: \Omega_0(M) \to \mathbb{R}$ (here $\Omega_0(M)$ is the space of contractible loops) by $$A_H(x)=\int_D u^*\omega+\int_0^1H(x(t),t)dt,$$ where $u$ is a capping disk for $x$ (we put an asphericity condition so that this is well-defined). One can prove that the critical points correspond to solutions of $\dot{x}(t)=X_t(x(t))$ (where $X_t$ is the Hamiltonian vector field), and that the flow of $-\nabla A_h$ is given by solutions of the Floer equation (disregarding the fact that the flow may not be globally defined etc).
A non-degenerate critical point is a path $x$ which is a critical point and such that the linearized flow has no eigenvalue $1$. The book by Audin, Damian claims that if all critical points are non-degenerate, then there are finitely many, but I don't understand why that is true. The question then is: Why nondegeneracy in this case implies finitely many critical points?
As a matter of fact, this has nothing to do with symplectic structures. The relevant fact is the following.
Lemma: Let $M$ be a compact smooth manifold, and let $\varphi:M\to M$ be a diffeomorphism whose fixed points are all non-degenerate. Then $\varphi$ has finitely many fixed points.
This lemma follows from the fact that a non-degenerate fixed point has to be isolated.
I think that the claim in your post does not necessarily hold when omitting the compactness assumption.