I believe the answer is positive if $M$ and $N$ are connected at infinity. Stein manifolds admit proper embeddings on vector spaces. An isomorphism from $M$ to $N$ can be represented by a collection of holomorphic functions from $M$ to $\mathbb C$. Each one of these extends to the whole $M$ according to Hartogs. Thus the isomorphism at infinity extends to a holomorphic map. Arguing in the same way with the inverse of the isomorphism at infinity yields an affirmative answer.

I believe the answer is positive in dimension at least two. Stein manifolds admit proper embeddings on vector spaces. An isomorphism from $M$ to $N$ can be represented by a collection of holomorphic functions from $M$ to $\mathbb C$. Each one of these extends to the whole $M$ according to Hartogs. Thus the isomorphism at infinity extends to a holomorphic map. Arguing in the same way with the inverse of the isomorphism at infinity yields an affirmative answer.