As Joel remarked in his comment, the answer is no when $\dim D =1$.
However, the answer is yes when $\dim D = 2$: see
B. Iversen, Numerical Invariants of Multiple Planes, American Journal of Mathematics 92, No. 4, (1970), in particular page 981.
More precisely, the following is true: if $\f \colon C \to S$ is a finite, flat cover of a smooth algebraic surface $S$ with branch locus $D$ and $x \in D$ is a smooth point of $D$, then any $x \in f^{-1}(x)$ is a smooth point of $C$.
It seems to me that Iversen's proof can be extended in any dimension $\geq 3$, although I did not check this carefully.