2 added 155 characters in body

The answer is affirmative when $X_K$ is smooth and geometrically connected with genus $\ge 2$ (which I am guessing are implicit hypotheses), without restriction on the residual characteristic nor on the generic characteristic. The key ingredients are theorems of Grothendieck concerning semistable reduction and the link between stable reduction of curves and semistable reduction of Jacobians (proved by Deligne and Mumford).

By increasing $K$ if necessary, the "minimality" hypothesis on $L$ reduces the problem to showing that if the ${\rm{Gal}}(L/K)$-action on the special fiber $X_k$ of $X_{R_L}$ is trivial then $X$ has stable reduction over $R$. By Deligne-Mumford, it is equivalent to deduce that the Jacobian $J$ of $X_K$ has semistable reduction. By Grothendieck's inertial criterion for semistable reduction, this in turn is equivalent to deducing that the action of $G_K = {\rm{Gal}}(K_s/K)$ on $V_{\ell}(J)$ is unipotent for some (equivalently, any) prime $\ell \ne {\rm{char}}(k)$.

By a theorem of Raynaud, $P := {\rm{Pic}}^0_{X_{R_L}/R_L}$ is a scheme, and more specifically is semi-abelian with generic fiber $J_L$. The subgroup $G_L = {\rm{Gal}}(K_s/L)$ of $G_K$ acts unipotently , on $V_{\ell}(J)$, and Grothendieck's orthogonality theorem (applied over $R_L$) gives much more: $V_{\ell}(J)^{G_L} = V_{\ell}(P_k)$ with $V_{\ell}(J)/V_{\ell}(J)^{G_L}$ canonically dual to $V_{\ell}(T)$ where $T$ is the maximal torus of $P_k$ (here using the auto-duality of $J$ over $K$). This "canonicity" includes $G_K$-equivariance, so to prove the unipotence of the $G_K$-action on $V_{\ell}(J)$ it suffices to prove the triviality of the ${\rm{Gal}}(L/K)$-action on $V_{\ell}(P_k)$. But this latter Galois action is just the composition of functoriality of $V_{\ell}$ applied to the ${\rm{Gal}}(L/K)$-action on $P_k = {\rm{Pic}}^0_{X_k/k}$ arising from the ${\rm{Gal}}(L/K)$-action on $X_k$. (Various implicit compatibility verifications are left as an exercise.) This latter action is assumed to be trivial, so we are done.

The answer is affirmative when $X_K$ is smooth and geometrically connected with genus $\ge 2$ (which I am guessing are implicit hypotheses), without restriction on the residual characteristic nor on the generic characteristic. The key ingredients are theorems of Grothendieck concerning semistable reduction and the link between stable reduction of curves and semistable reduction of Jacobians (proved by Deligne and Mumford).
By increasing $K$ if necessary, the "minimality" hypothesis on $L$ reduces the problem to showing that if the ${\rm{Gal}}(L/K)$-action on the special fiber $X_k$ of $X_{R_L}$ is trivial then $X$ has stable reduction over $R$. By Deligne-Mumford, it is equivalent to deduce that the Jacobian $J$ of $X_K$ has semistable reduction. By Grothendieck's inertial criterion for semistable reduction, this in turn is equivalent to deducing that the action of $G_K = {\rm{Gal}}(K_s/K)$ on $V_{\ell}(J)$ is unipotent for some (equivalently, any) prime $\ell \ne {\rm{char}}(k)$.
The subgroup $G_L = {\rm{Gal}}(K_s/L)$ of $G_K$ acts unipotently, and Grothendieck's orthogonality theorem (applied over $R_L$) gives much more: $V_{\ell}(J)^{G_L} = V_{\ell}(P_k)$ with $V_{\ell}(J)/V_{\ell}(J)^{G_L}$ canonically dual to $V_{\ell}(T)$ where $T$ is the maximal torus of $P_k$ (here using the auto-duality of $J$ over $K$). This "canonicity" includes $G_K$-equivariance, so to prove the unipotence of the $G_K$-action on $V_{\ell}(J)$ it suffices to prove the triviality of the ${\rm{Gal}}(L/K)$-action on $V_{\ell}(P_k)$. But this latter Galois action is just the composition of functoriality of $V_{\ell}$ applied to the ${\rm{Gal}}(L/K)$-action on $P_k = {\rm{Pic}}^0_{X_k/k}$ arising from the ${\rm{Gal}}(L/K)$-action on $X_k$. (Various implicit compatibility verifications are left as an exercise.) This latter action is assumed to be trivial, so we are done.