My Honda Accord is making whining on start-up

If the fluid was low enough to cause whining noises from the transmission, that is a fairly ominous sign.
Yes, keep checking the level and replenishing as necessary, but I would advise that you start putting money aside for a transmission overhaul, just in case my suspicions regarding transmission damage turn out to be true.

In the short term, as Volvo V70 advised, it would be a good idea to replace the battery soon, rather than having to incur the cost and the inconvenience of a tow.