How do you think Batwoman will react to meeting Supergirl first time?

In Batman and Superman's first meeting in the post-Crisis The Man of Steel, Superman came to Gotham to arrest Batman for being a vigilante. They eventually came to a meeting of the minds, but I wouldn't be surprised if something similar happened here.... particularly since Kara has already voiced her opinion on vigilantes in general.

They're not spelled Bat woman or Super Girl. They're spelled Batwoman and Supergirl. Please learn proper grammar.

In one of the panels at comic con Executive producer of Supergirl Jessica Queller said that strong women would be drawn to each other in a respectful way. That said, Kara’s power is unfathomable and I think Kate would look to someone she trusts to get a read on Supergirl. Absent that, she might keep low and try to learn a weakness as any Bat character is always looking for the upper hand.