This is scene-setting, with a bit of metonymy thrown into the description.
The Park is a place to the west, and this noun is used here to stand in for the things contained in the Park (trees, bushes, etc.) Everything in the Park is dripping, due to the weather, and so the metonymous phrase is used that the Park itself is dripping. The word "wretchedly" is used in sense 1 here, with a hint of sense 5:
wretched [...]
- Characterized by or feeling deep affliction or distress; very miserable.
Synonyms: dejected, (archaic or obsolete) wretchful
I felt wretched after my wife died.
[...]
- Of weather: causing much discomfort; very unpleasant; miserable.
Derived terms
So you could understand "the Park dripped wretchedly" as "the things in the Park dripped miserably".
The "great railway terminus" is also used metonymously, to represent the things contained in the terminus - trains and other noisy contraptions, the things that "slammed and banged and exploded hollowly". Remember that, at the time this novel is set, they would be steam trains, often a lot more noisy than the modern diesel or electric trains.
As for "about its affairs", this could be seen as personification. Usually, people are said to go about their affairs, but in this sentence, it's a railway station.
So you could understand "the great railway terminus slammed and banged and exploded hollowly about its affairs" as something like "the machines in the great railway terminus emitted slamming and banging and hollow exploding noises, as business went on as usual".