We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset LP relaxation for this problem. For directed graphs and $k$-out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node $v$ in the solution is at most $2b(v) + O(k)$ where $b(v)$ is the degree upper bound on $v$. For undirected graphs and element-connectivity requirements with maximum connectivity requirement $k$, our algorithm computes a solution that is a $4$-approximation on the cost, and the degree of each node $v$ in the solution is at most $4b(v)+O(k)$. These ratios improve the previous $O(\log k)$-approximation on the cost and $O(2^k b(v))$ approximation on the degrees. Our algorithms can be used to improve approximation ratios for other node-connectivity problems such as undirected $k$-out-connectivity, directed and undirected $k$-connectivity, and undirected rooted $k$-connectivity and subset $k$-connectivity.