The short answer is - very very small amount of internet bandwidth. Around 5mbps would work for even medium to large mining farm.
Red Panda Mining made a great video testing himself how much bandwidth a standard 6 x RX 580 Mining Rig consumes in one hour. He then tried a few different miners to compare them and see the difference.
I was focusing on his test on NiceHash, because I personally use it, and it's a great easy way to start mining.
In his test, mining with 6 x RX 580 graphics cards, he only used 89.6kb download and 66.2kb upload per hour. That is a very very small amount. Today's speed are 100mbps and up, which is thousands of times more than what he uses.
I want to point out latency though. Latency is how quick are you sending and receiving bytes over the internet. Not how many - which is bandwidth, but instead, how long does it take to send and receive data.
I'm mentioning latency because WIFI has higher latency than direct Ethernet connection. This effect your bottom line, and I even saw up to 10% difference in the mining speed. Therefore, I recommend connecting miners directly to your router for optimal mining profits. This applies to all miners, ASIC and GPU and CPU mining.
Also, in NiceHash, there is a way to change the server location that you are using for mining. It is very important that you choose the closest location to you in NiceHash, or any other pool of your choice. This will for sure increase your mining income.
So, in conclusion - if you plan to start mining in your home / office, do not upgrade your internet speed. If you have dedicated location where you only do mining - pay for the slowest cheapest speed available. Make sure to use wired internet connection using an Ethernet cable whenever possible. And choose a pool location as close to you as possible - in return you will maximize your profits. You can always buy digital currency as an alternative to mining.
Source - RedPandaMining - https://www.youtube.com/watch?v=AGBODtRV3zI