Question posed per the title.
Does the Alarm spell go off when invisible creatures enter the area? I would assume it would, as it makes specific caveats for ethereal or astral creatures and would have made one for invisible ones if applicable.
Alarm doesn't say anything about the intruder being visible, just that they enter the area. Invisibility won't stop the alarm from sounding, but will only make it harder for you to find the intruder.