It's the border with Austria near Munich that's being closed off. Not the whole country. I can't speak from experience but I'd be amazed if any airport other than Munich (and I don't believe that even this will happen) would be affected. It's road and rail that are the real issues here. Migrants are not arriving by plane. And it will be police and army more so than the passport booth lads and lassies that will be involved.
I'd be amazed if there was an issue, even more so on the basis that this is Germany we're talking about.