Is there constant current flow in this circuit?
Yes, you are thinking correctly. It will draw some minute current, even when 'off'.
The quiescent current would be that current going through the base bias resistors in the 'off' state. It would be very small when 'off', and while the bias current would drop slightly when 'on', the total current would be more with it on; (I know you knew that part!).
That would probably be a common cadmium sulfide cell (photo-cell), which I recall are about 100KΩ to 500KΩ in the dark, depending on it's specs; and a few kohms or so when exposed to moderate light. So it would be 100kΩ plus a few kΩs drawing current when exposed (transistor is off).
When 'on', it would draw roughly this: take the battery voltage, reduce that by .3V for the transistor if saturated* (saturated drop) and by whatever the LED will drop when lit (V-forward). a red LED may drop about 1.2V IIRC. That gives you 1.5V left over across the resistor. So divide 1.5V remaining by 330Ω and you get about 5mA* of current to light the LED.
The 'off' quiescent current would be about 3V ÷ ~100kΩ = 30µA. Very small as ambitiose_sed_ineptum aptly pointed out above!
*EDIT: td127 spotted a big error I made - Thanks td127!
The 2N3904 will not be fully turned on by this choice of bias resistor values. As td127 correctly commented earlier, (and I incorrectly contradicted him/her!) - the 2N3904 could have a gain of 100 or even less.
If you use a different transistor with adequate gain to saturate (fully turn on) at 30µA, then you should get about 5mA to the LED using the 330Ω current limiting resistor. You could also decrease the 100kΩ bias resistor (to provide enough base current to saturate a 2N3904), but that will also increase the 'standby' current some. That will also affect the 'turn-on' point/sensitivity.
I think the comments above already pretty much answered it all, but I put it in a formal answer anyway, with (excess) details, for other readers to understand all aspects of current flow in this circuit, which the question eluded to...