With Gigabit, the ONT and router must be connected using their Ethernet ports.
It would be easiest to simply add one of the MoCA devices linked above. You'll get a few hundred Mbps in the basement. WiFi, too, if you select the extender.
Another option is to use two MoCA 2.0 Bonded adapters to run the ONT<->Router WAN link over coax, but that coax can't be connected to any other devices. This is because the WAN link from the ONT can't be shared with anything else. If you have a dedicated coax available between the ONT and router, you could consider this. The ONT's and router's coax ports will still need to be connected to the coax network with the set-top-box for MoCA LAN and linear TV. This configuration is more complicated and higher cost.