If $f:X\to Y$ is a proper morphism of noetherian shemes, then there is a Stein factorization of the form
$$ X \xrightarrow{f'} Z={\bf Spec} (f_* O_X) \xrightarrow{g} Y $$where $g$ is finite and $f'$ has connected fibers. Furthermore $g_*O_Z=O_Y$ and ${f'}_*O_X=O_Z$. It follows that $f_*O_X=O_Y$ if and only if $f$ has connected fibers. So for example, cyclic covers do not satisfy this condition.