well you could leave it on all the time and wait for a trigger (the doorbell) thats a simple matter of software and some basic hardware to convert the doorbell voltages (what 12-24 vac?) to arduino happy signals.
a more economical approach is to put the arduino to sleep while it sits there doing nothing, dropping current draw from tens of millamps to microamps for the majority of the time, but its not exactly a direct function of the arduino library so you will have to get into some more advanced programming (though there are plenty of examples around)
either way the arduino consumes such little power by itself it shouldnt be a big deal, but going from 0.0x amps to 0.00x amps for something that will mostly sit there doing nothing does have its advantages as well.
which sounds better?