Let $X$ be a smooth surface of nonnegative Kodaira dimension. Then there exists a unique minimal model $S$ of $X$ where $K_S$ is nef, and $S$ can be constructed from iterated blow downs of (-1)-curves.
So, I think if $X$ is not minimal, then $K_X$ is not nef, and hence there exists a (-1)-curve on $X$. But I have no idea to prove this. If $K_X$ is not nef, then there exists an integral curve $C$ with $K_X.C<0$. But, how can I find a curve $C$ on $X$ such that $K_X.C=-1$?
We need an auxiliary lemma first.
Lemma:
Let $X$ be a surface and $C$ a curve with $K_X.C<0$. If $K\geq 0$ ($K$ being the Kodaira dimension of $X$), then $C^2 <0$.
Proof:
If $K\geq 0$, then $\mathcal O_X(nK_X)$ has a section for sufficient big $n$. In particular there exists an effective Divisor $D$ s.t. $D\sim nK_X$. Now we have $D.C=nK_X.C<0$. But $D$ was effective and as $C$ has positive or zero intersection with any other curve different from itself, we conclude $C^2<0$.
Proof of the statement:
Let $K_X$ and $C$ be as in the statement. Then by adjunction formula we have $-2\leq 2-g_C=deg(K_C)=K_X.C+C^2$. But by assumption $K_X.C<0$ and by the auxiliary lemma $C^2<0$. We conclude that $C^2=-1$, $K_X.C=-1$ and $g_C=0$. Thus we have found a curve with self-intersection $(-1)$ and which is isomorphic to $\mathbb{P}_1$ (since $g_C=0$), i.e. a $(-1)$-curve.