If you just connect an output of your 7404 chip to a BNC connector you don't have to worry about pull-ups or pull-downs on the output side. It looks like outputs of this chip cannot be tristated so you should always have a known output of the BNC connector. The output will be determined by the input of the NOT gate. You may need to put a pull-up or a pull-down on the input, but that depends on that portion of the circuit.
These mating BNC connectors (that would be inputs) however should probably have a pull-up or a pull-down. Which direction you pull depends on your circuit that mates to this circuit. Just think of the situation where you would have your BNC unconnected, resulting in an input that is not actively driven, do you want it to be GND or VCC?
I hope I'm helping and not confusing the situation more.