When one in NYC refers to "Brooklyn" especially in the late 19th century, they are referring to downtown Brooklyn, just off the Brooklyn Bridge. In New York City, the boroughs are made of small towns, so when they talk of where a New Yorker is from, they say "Flushing" or Brighton Beach" etc. So the character referring to Brooklyn as implying being far, it is a typical New Yorker way, and the area they speak of is about 10 miles, pretty far for horse and buggy or steam train.