I have found the following fact stated in a number of places:

If $k$ is any field, a connected reductive group $G$ is anisotropic if and only if its only unipotent element is $e$ and $\mathrm{Hom}_k(G, \mathrm{G}_m)$ is trivial.

For instance, this appears in section 3.4 of Springer's Corvallis article. However, I have been unable to track down a reference for a proof of this result.