When simmering broth should the lid on on or off?

Its usually 'off' so you can keep an eye on the simmering, If the lid  is on the liquid would boil over / overcook. 

Depends. 

If you are trying to reduce it, lid off.

I often do long, slow simmers or braises overnight while I am sleeping.  When I do that I have the lid on so the pan doesn't run out of moisture while I am asleep.

If I'm up and keeping and eye on it, I have the  lid off.

When there is a lid on the pot the temp inside the pot will go much higher because the heat is held in.  Depending on what I am doing I may want a simmer that is just barely bubbling, that can be hard to do lid on so I will get it going and check it multiple times before going to bed to be sure the temp won't be too high.

If I want a nice, clear stock or broth I want it to be just under a real simmer and let it go for a very long time.  If the solid contents are being moved around by an active simmer you'll get a murky broth.  For some things the level of translucency of the broth aren't important, but I tend to make broth long before I know what it's going to be used in so I go for a nice clear broth and very long low simmer.

I cover it to prevent evaporation.

I always simmer broth with the pan cover off 

No right or wrong answer to this.

Lid off means the liquid will boil away and reduce

Lid on and you maintain the liquid.

So, it really depends on what your cooking

Kind of depends on the dish. If you want to thicken the liquid slightly, leave it off. If not, keep all that heat in and cover it. 

On. You waste heat with the lid off. Only if you wish to reduce (thicken) a sauce should you simmer without a lid.