I know what you mean about hostels not being a particularly good deal for families. While under 20 Euros for a single is a good deal, when traveling with six you can often find a lot of good alternatives for under 120 Euros (at least during shoulder and low season).
Family rooms are like hotel rooms, you will have a lock and key for your room. Unlike hotel rooms, they will be more like sleeping over in a kids room with a collection of single beds. Usually a family room will have a dedicated bath room.
At Vondelpark hostel in Amsterdam, my family of five stayed in a room with six beds. They said they would try to keep it as a family room but when we arrived there was already a sixth person in the room.
We stayed at institute San Sebastian in Salzburg. They had planned on giving us adjoining three-bed rooms for our family. Each had a locking door. Outside the two doors was a bathroom in a small hallway with another locking door that seperated the two rooms and bathroom from the rest of the building. Unfortunately, when we arrived one of the rooms was occupied even though it had been reserved for us. We only found out after they gave us keys to both rooms. We could easily have taken anything of value in the occupied room if we were dishonest.
Even if they don't mess up your room at the hostel, they might still accidentally give a key to another person. I would keep my stuff locked up just in case. Bring one or two gym-type locks and keep your valuables locked up when you aren't physically present.
On the other side, we stayed in a great room with eight beds and a view at Burg Stahleck (the Rhine Blick room). They blocked the room off for just our family and didn't send in any surprise visitors.