By "enemies of the Union" the Herald means, of course, insanely radical abolitionist Republicans.
In the Herald's view, they should be given no recognition, sympathy or support whatever.
They are the people, so the Herald believes, who will destroy the Union.
And to this very day, that's just how Democrats continue to speak -- they take their own very worst sins (in this case, the threat of disunion), and accuse Republicans of somehow advocating it.
It's like Obama, Pelosi & Reed accusing Republicans of "shutting down the government", when the decision to do so is 100% their own!
Well put.
It was strange that the merged Herald-Tribune became an important voice in the Republican party since Bennett of the Herald and Greeley of the Tribune were opposite in politics (and in so many other things).
Bennett was very pro-South and violently anti-Lincoln, but had to temper his views once the war started. Bennett was erratic, though, supporting Fremont in 1856 and Breckenridge in 1860. More here and here.