I don't know of anyone successfully using satellite dishes for internet from a boat, but there might be such connections on cruise ships.
What I do is connect by cell phone. Many cell phones will look like a modem to the computer, and the connection is a normal phone call as far as the cell provider is concerned. So I do most of my stuff during free nights and weekends. The only problem is that the speed is only about 14.4. Some carriers offer new technology with much higher speed, but at a pretty high cost (like hundreds per month).
About 20% of the time as I've been cruising up and down the coast, I'm not in an area with good coverage. These spots are usually far from any signs of civilization, so poor coverage is not surprizing.
Most marinas provide a free phone line. In uncrowded marinas, that works well. Where I am now, when I dinghy in to the marina, there are always people in line for the phone line, and the limit is 15 minutes per use. (I'm not at dock, but out in the river at anchor. I pay the marina a small daily rate to anchor here and use their facilities.)
A lot of people who live on their boats don't travel, but either stay in the same place all the time, or spend half the year in one place and the other half the year in another. They often get phone lines to their dock.
Since we travel for weeks then stop for weeks, we have to rely on the cell phone and/or marina line.
The access is dramatically better this year than it was last year. I expect it will be even better next year.
I'd say that about 25% of the people I meet on the water use cell phones connected to their laptops, as I do for my primary access. Perhaps 5% connect for email only via amateur radio. Probably 50% use "Pocketmail", a service and device using a specialized PDA which has a built-in acoustic modem that can be held up to the handset of a payphone for sending and receiving email.
Most seem to have multiple ways to connect, as some are faster (cell phone) but don't work in the Bahamas without costly fees, or at all in some remote areas.
Probably 30-40% use marina lines only. 10% seem to use public libraries where ever they happen to stop.
We've done all of the above, but the cell phone connected to the laptop seems best all around.
Dory36, via cell phone
Do you ever get up to the Penobscot Bay, Maine? My parents live on Islesboro, ME. And that's where I'll be the first couple of weeks in February (hey! I know it's cold, but there aren't any tourists! And I don't have to fight for a seat at Moody's Diner). 
