While sailing on our transatlantic cruise on the Grandeur of the Seas, we had internet access through the ship's Wi-Fi access, although it wasn't cheap. We had the option of paying $0.65/minute or purchasing a pre-paid Internet package:
The Grandeur WiFi Hot Spot Locations include
While on the ship we typically used the Wi-Fi in the Solarium or the Library. It worked well, but was slow - more like modem speed than high speed internet.
We also took advantage of Wi-Fi while in ports and regularly took our laptop with us when we left the ship so we could grab a drink and check email before we left port.
The previous post in this blog was Cruise Ship Itinerary Day 2.