Is action maximized for a system in stable equilibrium?

We assume OP is investigating an action functional of the form $$S~=~int_{t_i}^{t_f}! dt ~L, qquad L~=~T(q,dot{q})-V(q),$$ with boundary conditions (BCs).

The classical solution(s) (i.e. solutions to the EL equations satisfying the BCs) do not necessarily minimize/maximize the action, cf. e.g. this Phys.SE post.

For generic BCs, a solution will not be static, i.e. a constant solution at a stationary point $q_0$ of the potential $V$. In order to have a static solution, the BCs should be Dirichlet BCs $$ q^j(t_i)~=~q^j_0~=~q^j(t_f), $$ where $q^j_0$ is a stationary points of the potential $V$. Then the stabile solutions are minimum points for the potential $V$. However, these are not necessarily maximum path for the action $S$, cf. pt. 2 and OP's title question.

Finally let us assume that the kinetic term $T$ is absent even off-shell, i.e. the Lagrangian $L=-V$ is static. (Consider e.g. non-relativistic massless point particles.) Then BCs would over-constrain the system, so we should also remove the BCs to be consistent. Then the classical solutions will be static solutions. Then the stabile solutions are minimum (maximum) points for the potential $V$ (Lagrangian $L$), respectively, cf. OP's title question.