The answer is yes when $\dim D \leq 2$ and the variety upstairs (i.e. $C$ in your notation) is normal: see

Bas Edixhoven, Robin de Jong, Jan Schepers, Covers of surfaces with fixed branch locus, Lemma 2.1

for a proof in dimension $2$ (which also works in dimension $1$).

It seems plausible that this proof can be extended in any dimension $\geq 3$, although I did not check it carefully.

The answer is yes when $\dim D=2$ and the variety upstairs (i.e. $C$ in your notation) is normal: see

Bas Edixhoven, Robin de Jong, Jan Schepers, Covers of surfaces with fixed branch locus, Lemma 2.1.

It seems plausible that this proof can be extended in any dimension $\geq 3$, although I did not check it carefully.

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.

The answer is yes when $\dim D \leq 2$ and the variety upstairs (i.e. $C$ in your notation) is normal: see

Bas Edixhoven, Robin de Jong, Jan Schepers, Covers of surfaces with fixed branch locus, Lemma 2.1

for a proof in dimension $2$ (which also works in dimension $1$).

It seems plausible that this proof can be extended in any dimension $\geq 3$, although I did not check it carefully.