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*][1], 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.   



  [1]: http://www.jstor.org/discover/10.2307/2373405?uid=3738296&uid=2129&uid=2&uid=70&uid=4&sid=21102178180061