There is a formula, $K_X\simeq \mathcal{O}_X(-2)\otimes \pi^*\det \mathcal{E}^{\vee}\otimes \pi^*K_{\mathbb{P}^n}$.

(More generally, for rank $r+1$ bundle, replace -2 by $-r-1$ in the above formula.)

Now $\pi^*K_{\mathbb{P}^n}\simeq \pi^*\mathcal{O}_{\mathbb{P}^n}(-n-1).$
So
$$K_X\simeq \mathcal{O}_X(-2)\otimes \pi^*\det \mathcal{E}^{\vee}\otimes\pi^*\mathcal{O}_{\mathbb{P}^n}(-n-1)=\mathcal{O}_X(-2)\otimes \pi^*(\det \mathcal{E}^{\vee}\otimes\mathcal{O}_{\mathbb{P}^n}(-n-1))$$

Hence $$-K_X\simeq \mathcal{O}_X(2)\otimes \pi^*(\det \mathcal{E}\otimes\mathcal{O}_{\mathbb{P}^n}(n+1))$$

So $-K_X$ is always effective as long as the line bundle on the RHS above has a non-zero section.

**Added** (I will leave this here for future readers): When I wrote this, I wasn't aware of Hartshorne's conjecture for rank 2 bundles. Thanks to @YosemiteStan for the enlightenment! See YosemiteStan's and abx's comment below for more details.