By a theorem of Miyaoka (Theorem 6.1 in Y. Miyaoka `[The Chern classes and Kodaira dimension of an algebraic variety][1]', 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][2], and later [Peternell][3].

See also Theorem 1.2 in Q. Xie `[On Pseudo-Effectivity of the Second Chern Classes for Terminal Threefolds][4]'.


  [1]: http://www.mast.queensu.ca/~mikeroth/NotesForSeminars/Miyaoka-Chern-classes-and-Kodaira-dimension.pdf
  [2]: https://projecteuclid.org/euclid.pja/1148393517
  [3]: https://eudml.org/doc/277496
  [4]: https://arxiv.org/pdf/math/0409269.pdf