Here's a counterexample with $A$ and $B$ of the same Krull dimension. Take $A = B = \mathbb{C}[x,y]_{(x,y)}$, and let $f: A \to B$ be the composition of the quotient of $A$ by $(y)$ with the embedding $\mathbb{C}[x]_{(x)} \to \mathbb{C}[x,y]_{(x,y)}$. $A$ is a regular local ring, so by the Auslander-Buchsbaum theorem $B$ has finite projective dimension as an $A$-module. However, the map cannot satisfy the going-down property because the induced map $f^*:$ Spec $B \to $ Spec $A$ is not surjective. Geometrically, $f^*$ projects the plane onto the $x$-axis, then embeds the $x$-axis into another plane (with everything localized at the origin). Clearly its image is the closure of the $x$-axis in the codomain. 

For a counterexample with the map $f: A \to B$ additionally required to be injective, take $A = \mathbb{C}[x,y,z]_{(x,y,z)}, B = \mathbb{C}[u,v,w]_{(u,v,w)}$ and let $f$ send $x, y, z \mapsto u, uv, uv^2$. Once again Auslander-Buchsbaum shows this has finite projective dimension, but going down fails: the prime $(x,y,z)$ lies under $(u, v)$, but the former has height 3 in $A$ and the latter has height $2$ in $B$.