  We consider the finite $W$-superalgebra $U(\mathfrak{g_\bbf},e)$ for a basic
Lie superalgebra ${\ggg}_\bbf=(\ggg_\bbf)_\bz+(\ggg_\bbf)_\bo$ associated with
a nilpotent element $e\in (\ggg_\bbf)_{\bar0}$ both over the field of complex
numbers $\bbf=\mathbb{C}$ and over $\bbf={\bbk}$ an algebraically closed field
of positive characteristic. In this paper, we mainly present the PBW theorem
for $U({\ggg}_\bbf,e)$. Then the construction of $U({\ggg}_\bbf,e)$ can be
understood well, which in contrast with finite $W$-algebras, is divided into
two cases in virtue of the parity of
$\text{dim}\,\mathfrak{g_\bbf}(-1)_{\bar1}$. This observation will be a basis
of our sequent work on the dimensional lower bounds in the super Kac-Weisfeiler
property of modular representations of basic Lie superalgebras (cf.
\cite[\S7-\S9]{ZS}).
