For a non-negative integer $k$, a vertex cut in a graph is $k$-degenerate if it induces a $k$-degenerate subgraph. We show that a graph of order $n$ at least $2k+2$ without a $k$-degenerate cut has the size at least $\frac{1}{2}\left(k+Ω\left(\sqrt{k}\right)\right)n$ and that a graph of order $n$ at least $5$ without a $2$-degenerate cut has the size at least $\frac{27n-35}{10}$. For $k\geq 2$, we show that a connected graph $G$ of order $n$ at least $k+6$ and size $m$ at most $\frac{k+3}{2}n+\frac{k-1}{2}$ has a minimum $k$-degenerate cut.