I encountered this frustrating message in Chrome yesterday and got around it by simply attempting the same thing from Internet Explorer 10.
IE10 gives you the warning "Do you want to allow this webpage to access your Clipboard?" and if you click Allow Access, it pastes like you'd expect.
For now, I'd much prefer to use a clunky copy/paste process in a clunky browser than be required to install loads of Google binaries on my system.
If anyone from Google happens to be reading, can I point out that not all consumers want their Google-hosted data so tightly coupled to their hardware. The prospect of installing an application that sounds as though it's going to cache your credentials, then start rummaging around in your cloud data and presenting the contents to whoever happens to be sitting in front of your machine (or holding your tablet/phone, or watching your TV) at the time is not appealing.