There is probably a lot of Mavericks bugs it's a brand new update and a lot of the developers have yet to update their apps fixing issues. However this thread was created simply to inform the Evernote team of the issue. There might be some sort of fix that they can do or it might simply be a Mavericks bug. It's also not my job to do QA and I'm sure there is some very capable people already looking for these types of things.
This bug happens when you are using multiple monitors with Evernote in fullscreen. If you are active on the primary display and evernote is open in fullscreen on the secondary display and you were previously editing a note the blinking cursor remains there. This can be misleading to the user where the user believes they are active on Evernote on the secondary display when they are not. eg. A user is using Evernote (in fullscreen) on one display and using Safari on the other. When the user goes to edit the note the typing will actually be happening on the safari window and not Evernote like the user may think. This is just throwing me off thought I would let you know.