Serial Connection Not Working on Modified Router
Firstly, the Online / Offline indicator is a bit of a misnomer.  The online state is tied to the Carrier Detect (CD) line being asserted, and that is usually done when a real modem has negotiated a proper connection to a remote modem.  You don't have a CD line on a little USB dongle like that, so you can completely ignore the Offline status.

Secondly, it's quite common for the console to not give any output except during the first few moments of booting. If it doesn't get a signal (pressing a key at the right time) it will assume there is no console connected, and not bother wasting time outputting data to something that's not there.

By disconnecting the +3.3V pin and powering the board externally it will give you the ability to power-cycle the board.  By being connected at the time you power cycle the board you should get an initial boot message including something along the lines of "Press any key for console mode" or similar.  Pressing a key (or the right key if one is specified) should drop you into console mode, at which point you should be able to see the output you'd expect.