  Let $\mathfrak{g}=\mathfrak{g}_{\bar0}+\mathfrak{g}_{\bar1}$ be a basic
classical Lie superalgebra over $\mathbb{C}$, and
$e=e_{\theta}\in\mathfrak{g}_{\bar0}$ with $-\theta$ being a minimal root of
$\mathfrak{g}$. Set $U(\mathfrak{g},e)$ to be the minimal finite
$W$-superalgebras associated with the pair $(\mathfrak{g},e)$. In this paper we
study the highest weight theory for $U(\mathfrak{g},e)$, introduce the Verma
modules and give a complete isomorphism classification of finite-dimensional
irreducible modules, via the parameter set consisting of pairs of weights and
levels. Those Verma modules can be further described via parabolic induction
from Whittaker modules for $\mathfrak{osp}(1|2)$ or $\mathfrak{sl}(2)$
respectively, depending on the detecting parity of
$\textsf{r}:=\dim\mathfrak{g}(-1)_{\bar1}$. We then introduce and investigate
the BGG category $\mathcal{O}$ for $U(\mathfrak{g},e)$, establishing highest
weight theory, as a counterpart of the works for finite $W$-algebras by
Brundan-Goodwin-Kleshchev and Losev, respectively.
  In comparison with the non-super case, the significant difference here lies
in the situation when $\textsf{r}$ is odd, which is a completely new
phenomenon. The difficulty and complicated computation arise from there.
