How do systems take damage from Beam weapons?

Advanced Edition makes this much easier since it actually highlights the rooms that will be affected by the beam when you are aiming it.

Basically, the beam has to touch inside the walls of the room. It doesn't have to go inside by much, so you can often squeeze an extra room in if you are careful. But you can't put your beam parallel to the wall between two rooms and expect it to hit both rooms. For example, there is no way to aim the beam so that it hits both the shields and the large square room below the oxygen in the picture below.

Note that the Anti-Bio beam actually works differently. Even though it highlights rooms like a normal beam, it only affects crew members, and it only affects them if the path actually touches the square that the crew member is in. So in the screenshot above, my anti-bio beam actually wouldn't hurt anyone even though both rooms are highlighted, because it doesn't cross the actual crew members shown in red.