First all boats should have a switch between the battery and any of the positive bus bars, circuit breakers or engine cables.
I really recommend a second battery as a back up--and that way you can run the electronics without subjecting them to low voltage or high spikes.
Many items on a boat have a parasitic draw--radios and some GPS have memories, the bilge pump may be one which comes on every few minutes and checks for water. There can also be minimal shorts due to corrosion.
I certainly would test the battery--and might even go a bit further than Denny-o, by load testing it.
If it has been run down, less than 11 volts, it will damage the battery. I would buy a new battery, along with a switch, before even starting the tests.
A light will work to see if there is current flow, but with very low current it may be very dim or even not visiable. Try the light first in series with the positive lead. If that does not light up, I would then check with the volt meter. Most have a 10 amp circuit--separate from the low current flow which is measured in milliamps. You have to plug the red lead into the special socket. For example one in-expensive volt meter I carry in the boat is the Sperry DM 4100A You can find them for about $15, and the manual is at:
http://www.sperryinstruments.com/documents/products/dm-4100a.pdf
There are more expensive clamp on meters, but first using the light, and then this type of in expensive meter it will give you a good idea of how much current draw there is.