We were on the Explorer in June. I brought my laptop (which is already set up for wireless internet) and was quite glad I did. They have several packages available. The one I purchased was fifty dollars for which I received I believe 150 minutes. As Sonny said, there are several hotspots on the ship, one being the Cafe Promenade. While I understand the onboard computers are fairly "slow" the wireless was high speed. There is also another program wherein you can obtain service from your stateroom. I preferred the Wi-Fi, however, because for me it was the best bang for the buck.
You may wish to check out Tom Ogg's latest Mega Review for the Explorer for he goes into more detail. I have a review of our June cruise posted as well but I'm a mere novice while Tom is the expert.
Hope you have a grand cruise!