Floating pins though can cause power drain in sleep mode. Fortunately the 328 has some hysteresis in the input stages which may help.
Basically any "standard" CMOS input stage is an inverter and will consume non-trivial power if the input is somewhere in the middle of the range (the "forbidden region"). Here non-trivial means its an issue for battery-powered circuitry in sleep mode - something on the scale of 1mA might flow from that one input circuit when the whole chip is supposed to be shutdown (a few uA usually).
Also a floating input could be sent into oscillation due to capacitive feedback - in theory at least - symptom is increased power drain in that input circuit and gates further down the line.
In sleep mode the 328 and other AVR microcontrollers disconnect the input circuitry from the pins and grounds the input inverter, so none of this is an issue in sleep mode.
In general it is wise to avoid floating inputs in any deployed circuit - for breadboarding you'd only worry if it stopped things working.
For the 328 the advice to use internal pull ups is good I think, but few bother for prototypiing.