Our metro area just got a new airport this year, there are the usual men's/women's restrooms and non-gendered restrooms. I've been in both, it's really a non-issue, as long as you don't mind washing your hands next to a...gasp...member of the opposite gender
When I think about it, I wonder how we even got to the idea of separate restrooms--another era I suppose. FWIW--the stalls in the non-gendered restrooms have the European style stalls, there's almost no space at the bottom of partitions.
We like to say for anyone who finds the airport restrooms problematic, wait till you get on the plane!