By a theorem of Miyaoka (Theorem 6.1 in Y. Miyaoka `The Chern classes and Kodaira dimension of an algebraic variety', 1987), we have that for a Fano variety $X$, $c_2(X)\cdot H\ge 0$ for any ample divisor $H$. By duality of the cone of curves and the nef cone, it follows that $c_2(X)$ is at least pseudoeffective (i.e., a limit of effective classes). On the other hand, since $X$ is Fano, the Cone theorem shows that the cone of curves is rational polyhedral, spanned by finitely many classes of rational curves, so $c_2(X)$ is in fact effective.
EDIT: As C. Jiang correctly points out, to apply Miyaoka, one needs also the fact that the tangent bundle of a Fano 3-fold is generically nef. This was proved by Kollár--Miyaoka--Mori--Takagi, and later Peternell.
See also Theorem 1.2 in Q. Xie `On Pseudo-Effectivity of the Second Chern Classes for Terminal Threefolds'.