It’s been widely reported that Mavericks and Gmail don’t play well together. According to 9to5 Mac, Apple are working on a fix, but in the meantime, you might want to try Airmail.
Airmail is the top paid App in the Mac App store right now, and at only $1.99, it’s a great little client for Gmail. It’s interface is an unabashed homage to Sparrow, with new Mavericks-only features, such as replying to emails in the Notification Centre.
I personally prefer Gmail in a browser, but if you want a native App, Airmail is pretty great.