The Route of the Ring

[CSteefel]

The Wise certainly knew there was some "problem" in Moria, as did the Dwarves. I don't recall if there were any survivors when Durin VI was slain by a Balrog in 1980 of the 3rd Age, but they refer to threat simply as "Durin's Bane", rather than mentioning a Balrog specifically. And in the War of the Dwarves and Orcs, Dain Ironfoot ventured inside the Gates of Moria after slaying Azog, it was said that:

Yet hardy and full of wrath as he was, it is said that when he came down from the Gate he looked grey in the face, as one who has felt great fear.

And then,

But we will not enter Khazad-dum. You will not enter Khazad-dum. Only I have looked through the shadow of the Gate. Beyond the shadow it waits for you still: Durin's Bane. The world must change and some other power than ours must come before Durin's Folk walk again in Moria.

Presumably this is the sort of information that led Aragorn to counsel against entering Moria with the Fellowship of the Ring, even specifically referring to the threat to Gandalf:

It is not of the Ring, nor of us others that I am thinking now, but of you Gandalf. And I say to you: if you pass the doors of Moria, beware!

But it also seems clear enough that nobody knew it was a Balrog, as the quotes in the posts above indicate.