We study fractal properties of the image and the graph of Brownian motion in $\R^d$ with an arbitrary c{à}dl{à}g drift $f$. We prove that the Minkowski (box) dimension of both the image and the graph of $B+f$ over $A\subseteq [0,1]$ are a.s.\ constants. We then show that for all $d\geq 1$ the Minkowski dimension of $(B+f)(A)$ is at least the maximum of the Minkowski dimension of $f(A)$ and that of $B(A)$. We also prove analogous results for the graph. For linear Brownian motion, if the drift $f$ is continuous and $A=[0,1]$, then the corresponding inequality for the graph is actually an equality.