How to use a transistor to externally trigger a Canon EOS camera?
I am assuming that the Arduino acts only as a power source for the BJT's base. In that case, you need to complete the current path by connecting the BJT's emitter to Arduino's GND.
Also you should be aware that when the transistor is in its conductive state (eg. the button is pressed) it is able to conduct current only in one way (from collector to emitter). Assuming that you do not know the exact pinout of the camera jack (which of the two pins is ground) you may have to also swap the wires going to the camera.