When is the universal cover of a Riemannian manifold complete?

It's actually true that $M$ is complete if and only if its universal cover $\widetilde{M}$ is complete. Let $p: \widetilde{M} \to M$ be the universal covering map, $q \in M$ and $\tilde{q} \in p^{-1}(q)$. As has already been stated, by Hopf-Rinow, all we need to if we want to conclude that $\widetilde{M}$ implies $M$ is complete is prove the corresponding statement for the exponential maps based at $q$ and $\tilde{q}$.

Now if $\widetilde{M}$ is complete and $\widetilde{E}: T_{\widetilde{q}}\widetilde{M} \to \widetilde{M}$ is its exponential map based at $\widetilde{q}$, define a map

$$ E = p \circ \widetilde{E} \circ (dp)^{-1}: T_qM \to M$$

You can show that $p$ sends geodesics to geodesics by showing their images are locally length minimizing. Since $(dp)^{-1}$ is linear it sends radial lines to radial lines, and you can use this to show that $E$ is exactly the exponential map for $M$ based at $p$. Then from the above it follows immediately that $E$ is defined on the whole tangent space.

The other answer shows the reverse implication.