Watch for worn gearboxes. These were clunky little affairs from new, so don't worry if it is clunky, that is normal for ZZR600 boxes, and just keep an eye out for miss shifts into 2nd or 3rd.
As Smitty mentioned cam chain tensioners shit themsleves easily. However they are a 5 minute fix. Whereas a worn cam chain is a more involved job. At a real 37k, it could be on the way out anyhow.
Headstem bearings were commonly done on these too.
Other than that, I cannot say enough good things about Kawasaki's ZZR600 E models (1993 on). They are a fantastic 600cc bike. Many a vertically challenged girl has asked me what they are going to buy after they sell their ZZR600. My answer to date has always been, find another low kilometre one, and buy it!!!