For curves, bigness/ampleness/nefness is just a fancy way of asking whether the degree of the divisor is positive. The degree of $K_X$ is $2g_N-2$, and the degree of $D$ is equal to the number $c_N$ of cusps. So the question is equivalent to asking when $k(g_N - 1 + c_N/2) - c_N$ is positive.
Suppose that $N \ge 3$, to avoid issues with orbifolds. The Galois group $G$ of $X(N)$ over $X(1)$ is $G = \mathbf{SL}_2(\mathbf{Z}/N \mathbf{Z})/\{\pm 1\}.$ Since $X(1)$ has only one cusp, the group $G$ acts transitively on the cusps of $X(N)$. The stabilizer of $\infty$ just consists of the matrices $\displaystyle{\left(\begin{matrix} 1 & * \\ 0 & 1 \end{matrix} \right)}$, which is a group of order $N$. Hence, by the orbit-stabilizer theorem, $c_N = \frac{1}{N} \cdot |G|.$ Finally, by using the Riemann-Hurwitz theorem, the genus of $X(N)$ is $g_N = 1 + \frac{1}{12} |G| - \frac{c}{2}.$ (We use the fact that there are no elliptic points on $X(N)$, since $N \ge 3$.) Thus $ \deg(k L - D) = k(g_N - 1 + c_N/2) - c_N = |G| \left(\frac{k}{12} - \frac{1}{N} \right),$ which is positive if and only if $k > 12/N$.