What does "Lady Who Objects" mean?

Hard to know who it is specifically without context, but this person is someone who nags them. Could be their mother, could be the landlady, could be someone else who is watching them, could just be one of the group who always has a gripe.

Ferber's tale relates to the circumstances of New York working-class girls in about 1900 or thereabouts.

A fuller quote is:

All of those ladies who end their conversation with you by wearily suggesting that you go down to the basement to find what you seek, do not receive a meager seven dollars a week as a reward for their efforts. Neither are they all obliged to climb five weary flights of stairs to reach the dismal little court room which is their home, and there are several who need not walk thirty-three blocks to save carfare, only to spend wretched evenings washing out their handkerchiefs and stockings in the cracked little washbowl, while one ear is cocked for the stealthy tread of the Lady Who Objects.

American Literature

The first part "All of those ladies ... what you seek" refers to those who (we presume) live in relative comfort. The paragraph then goes on to compare the circumstances of others less fortunate: low paid women who only get $7 per week; who live high up stairs in apartment blocks; who cannot afford taxis and buses so walk long distances (33 blocks is roughly 2 or 3 miles); who do their small items of laundry in their washbowl. And the "Lady who objects" is likely to be their landlady, who objects to their using the apartment to make a little money, to their splashing water around, to their drying of clothes. It might also be a querulous and busybody neighbour who dislikes or disapproves of the activity.