Hola Aron,

Both sentences would be correct and expressing pretty much the same thing. As you said,  the natural placement of "menos" would be different when used with "cuanto más":

"Cuanto más vamos envejeciendo, menos nos van importando las cosas banales."

I have to say that even though both sentences mean the same, one is more focused on the idea of these two things happening at the same time, as parallel events:

"Conforme vamos envejeciendo, nos van importando menos las cosas banales."

And this one would be more focused on this idea:  "The older we get the less important things are.", so it is not necessarily seen as parallel actions.

"Cuanto más vamos envejeciendo, menos nos van importando las cosas banales."

But the difference is very very subtle.

Saludos.

Inma