Who Me Too'd this solution

If you want to leave the router on the first floor I would imagine you also have a coax wall plate there as well? Just connect the coax to the router and get a fios network adapter from verizon for $55. You can place it down in the basement and connect it to all the ethernet wiring in the house. The coax connected to the router will feed down to the panel and connect to the fna. The the fna has 4 ethernet ports with can feed the wall plates around the house. 
https://www.verizon.com/home/accessories/fios-network-adapter/