Some ships do have "Cyber Cabins" where you can get online in your room.
From the RCCL website:
"Is Internet access available from the ship?
Yes, royalcaribbean online allows guests to access e-mail and the Internet for a small fee. The following ships offer internet access in your stateroom using a Cyber Stateroom hookup that is available for rental while onboard: Adventue of the SeasŪ, Brilliance of the SeasŪ, Explorer of the SeasŪ, Jewel of the SeasŪ, Mariner of the SeasŪ, Navigator of the SeasSM, Radiance of the SeasŪ, Serenade of the SeasŪ, and Voyager of the SeasŪ.
Please note: We have a limited number of computer cables available for rental."
All others do have Internet Centers and Hot Spots where you can use your own laptop with a WiFi card. See also this thread for prices