In all of these examples, "down" and "down to" could not be exchanged.
"go down the mine" means descend into the hole in the ground.
"go down to the mine" means go along the street to the mine buildings.
You can say "go down to the shop" but in standard UK English you don't say "go down the shop".
"go down to the basement" means "go down the stairs until you reach the basement. In standard British English you don't say "go down the basement", since the basement is level.
"water dripped down the wall" means water descended in drips on the wall.
"water dripped down to the wall means water dripped down until it stopped at the wall.
In these cases "down X" means you remain in, or on X as you go down. But "down to X" means you go "down" until you reach X. Sometimes "down" is figurative. In some dialects the "to" can be dropped.