I was wondering if it is foolish to ask if there is a criteria on a morphism $f: X \to Y$ between separated schemes of finite type over a perfect field which will assure that all the scheme theoretic fibers over closed points of $Y$ are reduced. I can assume that the morphism $f$ is flat and finite.

As a commutative algebra question this becomes the following: Let $\phi: A \to B$ be a flat, finite morphism between finite type $k$-algebras over a perfect field. Is there a criteria on $\phi$ which will ensure that for every maximal ideal $m$ of $A$ the algebra $B / \phi(m)$ is reduced?