Text that displays the stair orientation isn't an available display option in the stair styles. So you are left with the choice of adding it manually as text objects or creating a tag that accesses the stair object's verticalorientation property and displays it as "UP" or "DN" or whatever. In order for the tag to work properly, you would need to:
1)Add the stairobjects property set, preferably by transferrring it from schedulestyles.dwg.
2)Add a handle property to that set.
3)Add a formula property with a meaningful name, like Orientation.
4)Add the property set to the stair.
5)Create a tag that displays the Orientation Property.
6)Add the tag to the stair object.
That's a lot of work and the main advantage would be that the relationship between the stair oriention and tag would be maintained. But how often will that ever change? In case you really want to go through the effort of tagging, the formula property would need to be coded something like this:
On Error Resume Next
Set app = GetObject(, "AutoCAD.Application")
Set doc = app.ActiveDocument
Set StairObject = doc.HandleToObject("[Handle]")
orient = StairObject.VerticalOrientation
If orient = 1 Then RESULT = "UP" Else RESULT = "DN"
StairOrientationTag Screencast