Fastest polynomial time algorithm for solving minimum cost maximum flow problems in bipartite graphs
See this paper. I think adopting their technique, many(usually scaling based) min-cost flow algorithms with running time $O(f(n,m))$ can run in $O(f(n_1,m))$ time. Here $n_1$ is the size of the smaller bipartition.

If $f(n,m)$ is the best possible run time for min-cost flow on general graphs with $n$ vertices and $m$ edges. There is an easy argument that you can't expect anything running in $o(f(n_1,m))$ time. (with some reasonable assumptions, like $n\leq m$, $f(n,cm)=O(f(n,m))$ for any constant $c$)

Proof: Otherwise, you can apply the standard vertex splitting operation to the input graph and use the bipartite min-cost flow algorithm and get a better running time. 

Finally, if you are restricting cost and capacities things, these scaling algorithms is still pretty much state of the art(difference is at most $\log c$ for w/e parameter $c$ you care about). You can decipher that Theorem 3.5 to see which one of them gives you the best result.