So, the left side of that Leviton box, with the pink wires, all appear to be telephone related.
The right side of the box, the one with the grey wires, are all your Ethernet (network) connections to the rest of your home.
You should have lights on the switch down there, so check that out - at least the 'power' light should be on.
So, you have your modem in your office and you are running one cat-5 wire to your basement to feed the rest of the house?
That is fine, I'm just trying to understand if that's what you are doing.
There is no reason to run the WAN to the switch.
Cable Modem (Fios, Comcast, Cox, etc.) -> LAN Port out to Network Switch -> To all rooms/devices as necessary
Most modern cable modems have several ports on them, but some do not... In fact, some have a built in router, and some do not.
So - I guess this is more of a networking question, so let's start there.
What kind of Modem are you using? Can you take photos of that one piece of hardware?
If it is a modem and does not have an internal router, then you will need to purchase a router so that your computers in your home can receive IP addresses and you can add that switch to your system.
This photo kind of shows what a setup would be... Instead of a network camera, you would have multiple PCs sitting at the end of your network switch, but the concept is the same.