The English is somewhat ambiguous.
It could mean 'We go [down the stairs] [to the basement]'. That is, we go down to the basement, using some stairs or other.
But it could also mean 'We go [down the stairs to the basement]'. That is, we go down the specific stairs that lead to the basement. (Whether we take these stairs all the way to the basement is another matter.)
How does Hungarian distinguish these two cases?
I guess 'downstairs' is close enough here that it should be accepted. However, there is a difference in English between 'down the stairs' and 'downstairs'. The latter is more like a direction rather than a specific reference to stairs. So, for example, I could take the elevator downstairs. my friend lives downstairs, etc.
"Downstairs" is a bit of a funny word, since it is rather a direction/location than an actual object. "Downstairs" just means "on a lower floor", whichout specifically referring to stairs. You could also take the elevator or a slide downstairs.
But the Hungarian sentence is specifically mentioning that we use the staircase by using lépcsőn.