Ample vector bundles on curves

Here's a partial answer. Suppose we're in characteristic 0 (Fujita would be assuming this), and that $rank(E)=2$. By cor 7.6 of Hartshorne's ample vector bundles paper, it suffices to check that $deg(E)>0$ and $deg(L)>0$ for ay quotient line bundle. From Riemann-Roch as in Piotr's comment, we get $$deg(E) + rank(E)(1-g) = h^0(E)\ge 0$$ which implies positivity of $deg(E)$. On a curve $H^1(E)=0$ implies the vanishing for any quotient bundle, and so in particular for $L$. Combing this with the above argument, gives $deg(L)>0$.

I think this can be pushed, but I'd better back to the less fun things that I'm supposed to be doing now.

This builds on the idea of Donu Arapura: We use the following criterion of Hartshorne (Thm. 2.4 in this article): $E$ is ample iff $Q$ has positive degree on $C$ for any quotient bundle $Q$ of $E$. Let $r=rank(E)$ and $s=rank(Q)$. Here

$$deg(Q)=s(g-1)+\chi(Q)> \chi(Q)$$

so it suffices to show that $\chi(Q)\ge 0$. But this is elementary, since $H^1(E)$ implies $H^1(Q)=0$ by the exact sequence $$ 0\to L \to E\to Q \to 0. $$