. . . . . . .
. . . . 1 . .
. . . . s . .
. . . . s . .
. . . . . . .

1-character on elevated terrain
s- stairs

My question is can the character place the barrier down the stairs?

Stairs are mainly a graphical decoration to show where you can change elevations if you can't fly or have another way to change elevations.

In your example, if 1 is on a "column" and the stairway/ladder line is between 1 and the first s, I don't see any problem with placing the barrier there since squares containing a stair or ladder count as clear terrain.

The 2012 Rulebook tells us:

The problem is that those squares aren't adjacent to each other. Barrier tokens need to be placed in adjacent squares.

Page 14 of the RB:

Part of a stairway or ladder is on terrain of a
lower elevation and part is on terrain of a higher
elevation. A character must pass through the
two squares of different elevation to change
the character’s elevation. On some maps, the 2
squares of different elevations have numbered
triangles where the number indicates the level of
elevation of each side of the red boundary line.
Characters occupying the two squares through
which a character can change elevations can make
close combat attacks against each other as if they
were adjacent. Stairs and ladders are clear terrain,
unless there are map boundary lines indicating
otherwise.


So characters can move through and attack through stairs/ladders, because these rules here say so, but otherwise you've got two squares that are on different levels of elevation; there are not adjacent. No Barrier tokens can be placed across them.

Likewise (and this came up on Heroclix online), no carrying a character and setting him/her/it on top of a stairwell with the carrier sitting on the stairs on the lower elevation. That's for the same "they're not on the same elevation" reason, right?

Right, same issue.