You actually have to log in to Connect at Sea to use it. I turned off background refresh on all apps, and turned off cellular data. I did not put my phone in airplane mode because I paid AT&T for their cruise package and I needed people to be able to reach me. I would log in to Connect at Sea, download my email and log out again. Then I would draft email responses offline, log back in and send them, then log out again. My husband and I went through under 1 gig in 7 days, even with some social media logins and quite a bit of email usage for work.
Depending on your cell carrier, you can get voice and text service on the boat. It is not cheap, but if you have a job or family that requires being in touch, it works well. AT&T was $60 for voice and text for a month.