Meaning of "beneath the dignity of the office"?

Solution 1:

To be literal, high is an adjective used of people, institutions and qualities that are worthy of respect, honour and admiration.

Conversely, we use “low” when the opposite of these things are merited: contempt, shame and disgust (or something like these).

The presidency is among the highest offices in the USA - arguably the highest,

Accordingly, the conduct of the holder of that office is expected to show the qualities that match that office (I decline to specify what they are, to keep out of politics.

If, Instead, a president should conduct him or herself in a manner that was (say) dishonourable or dishonest, his or her conduct could be said to fall below the high standard expected of a person holding high office.

The preposition merely continues the metaphor. As to the office holder named in the question, I could not possibly comment,

Solution 2:

We expect a general level of civility/decorum/honor in a position as important as the president, as they're our representative to the world. In addition, in general Christianity, 'good' is associated with 'up' (towards heaven), and 'bad' is associated with 'down' (towards hell). So to say that someone is 'beneath the dignity of the office/position' is to say the person's moral qualities aren't quite to the standard we expect from our elected leaders.