Neither NMEA 0183 or NMEA 2000 have a "box". The NMEA 2000 network is powered by yellow wire that is hooked into the backbone. This power wire supplies power to things on the network that are not directly powered from the battery. Often these are things such as sensors, GMI 10, the Fusion remotes, etc. The Garmin marine network however does have a hub that may have lights on it depending on the model. I do not believe the Garmin network is POE. These lights are similar to the lights on a home network hub and may be on to show traffic or just that an item is connected and online. Depending on how your boat was wired both the hub and/or the NMEA may be powered when the electronics (radio, plotter, radar) are turned at the breaker or when another breaker is turned on because your stereo or other networked things may use the Garmin hub.
The "box" should go off with one of your breakers. Watch the box and have someone switch off the breakers. AYBC standards are such that most everything on your boat should be controlled by a breaker or have a fuse and be wired from the battery (things such as a bilge pump). If the "box" is fused at the unit use a test light on it and see it it remains energized when all the breakers are off. If it remains on you should have a marine electrician look at it as it should go off with a breaker.
With a picture I'm sure someone could tell you exactly what is still on.