= Solution
By <Kodaira's lemma> for the <big divisor> $K_X$, choose an integer $q>0$, an <ample Cartier divisor> $A$, and an <effective divisor> $E$ with
$$
qK_X\sim A+E.
$$
If an integral curve $C$ is not a component of $E$, nonnegative intersections of distinct curves give
$$
qK_X\cdot C=A\cdot C+E\cdot C>0.
$$
Thus every curve with $K_X\cdot C<0$ is one of the finitely many components of $E$. In particular,
$$
\boxed{\operatorname{Neg}(K_X)\subseteq\operatorname{Supp}_{\mathrm{components}}(E),\quad\operatorname{Neg}(K_X)\text{ is finite}.}
$$
This is the surface case of <negative curves of a big real divisor lie in finitely many divisors>. The proof does not assert that every negative curve on $X$ belongs to this finite set; it concerns curves negative against this particular big canonical class.
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