  An $r$-uniform hypergraph ($r$-graph) is linear if any two edges intersect at
most one vertex. For a graph $F$, a hypergraph $H$ is Berge-$F$ if there is a
bijection $\phi:E(F)\rightarrow E(H)$ such that $e\subseteq \phi(e)$ for all
$e$ in $E(F)$. In this paper, a kind of stability result for Berge-$K_{3,t}$
linear $r$-graphs is established. Based on this stability result, an upper
bound for the linear Tur\'{a}n number of Berge-$K_{3,t}$ is determined. For an
$r$-graph $H$, let $\mathcal{A}(H)$ be the adjacency tensor of $H$. The
spectral radius of $H$ is the spectral radius of the tensor $\mathcal{A}(H)$.
Some bounds for the maximum spectral radius of connected Berge-$K_{3,t}$-free
linear $r$-graphs are obtained.
