Mail App is using internet connection to capacity (Retina Macbook Pro Early 2013)
Mail > Preferences > Accounts > Advanced

There is an option called "Keep copies of messages for offline viewing"
If you have that set to All messages and their attachments, then Mail will download your entire Gmail archive, which could be several GB.
I suggest setting that to All messages but omit attachments, then quitting Mail, then deleting ~/Library/Mail/V2 and relaunching Mail.  It will take several minutes to clean up and re-download all the messages, but this time without attachments.