You can run miners on dial-up or satellite connections, the worst that could happen is going to be stale shares and possibly more invalid and rejected, consistent internet is the big factor. You only need a web accessible device to connect to the miner to configure it and you'll be good to go for the most part. Don't forget to check up on your miner every now and then to make sure nothing is going wrong.
I've never run my mining farm on dialup - that MIGHT be pushing bandwidth limits a LITTLE for a largish farm but should be fine for just a few mining rigs.
I did run it for a few years on a very poor quality Virgin Mobile 3G cell connection, that hit 400kbps on a GOOD wee hours of the night and was usually more like 150-200, no issues.
I did run it for 2 years on an Exede sat connection (which was a MAJOR UPGRADE from the 3G except SOMETIMES on lag), the 650-700ms ALL THE TIME lag cost less than 1% in stale shares on Litecoin or Ethereum mining and wasn't noticeable on Bitcoin mining.
Exede was metered, but I normally used less than 100 megs a day and that was INCLUDING online gaming and web browsing, NOT just mining (Game PATCHES were sometimes an issue).
I never assumed we were talking about farms here, but I actually tried using a couple miners on Dial-Up and it worked just fine. The losses you'll incur running such connections are minor in most cases like you mentioned either way, and dial up CAN be had for free or very cheap in many areas in the US and possibly Canada.
I wonder if a rasberry pi is good enough as a computer. You guys already said the computer specs don't really matter, just the internet connection. I'll have to research on that later.
As long as you can use a browser or even SSH to whatever IP the miner is running on, or you can open CGminer/BFGminer to run a USB-connected miner, it will work. For Bitmain devices, remember that the default login is root:root.
