First off, you obviously have to assume something about $-D$ not being effective, because otherwise you could take a negative ample class. 

The cone of curves of a K3 surface is pretty well described in [this paper][1]. And there is a newer version of it that works in positive characteristic as well [here][2].

Here is what you get out of this:

 1. It is possible that there are no $(-2)$-curves on a K3 surface, but in this case for every divisor with $D^2\geq 0$ either $D$ or $-D$ is both nef and effective.
 2. If the Picard number is $2$, it is possible that there is only one $(-2)$-curve. In this case there are actually (effective) curves with non-negative and even positive self-intersection which are not nef. However, they are negative on the sole $(-2)$-curve. (I think I will leave this for the reader for now).
 3. In all other cases the $(-2)$-curves generate a cone which is dense in the cone of curves, so any divisor that is non-negative on the $(-2)$-curves is non-negative on every effective curve.

So, it actually looks like that what you want is true.


  [1]: http://link.springer.com/article/10.1007/BF01450509
  [2]: http://arxiv.org/abs/1309.5562