In Harper Lee's To Kill a Mockingbird, what does Aunt Alexandra mean when she tells Scout that "people who held such views were usually climbers"?

Alexandra thinks that people who worry about what other people have consider themselves better than others, and are therefore social climbers. 

The conversation about social climbers relates to Scout and Cecil Jacobs.  At the Halloween celebration, Cecil tells Scout not to bob for apples. 

His mother said he might catch something from everybody’s heads having been in the same tub. “Ain’t anything around town now to catch,” I protested. But Cecil said his mother said it was unsanitary to eat after folks. I later asked Aunt Alexandra about this, and she said people who held such views were usually climbers. (Ch. 28)

Aunt Alexandra is implying that Cecil Jacobs’s mother is looking around at other people and looking down on them.  She feels that Cecil Jacobs’s mother is lower on the social ladder, and wants to be higher.  This is why Alexandra calls her a climber.

Connecting bobbing for apples with social class demonstrates how class-conscious Alexandra is.  She is constantly evaluating people, to see who is better than others.  Alexandra, of course, considers herself better.  The Finches are an old established family.  She wanted Atticus to impress upon Scout and Jem that they are better because they are Finches.  He had a hard time with this, because it made him uncomfortable since he usually considered all people valuable.

Our father was actually fidgeting. “No, I just want to explain to you that—your Aunt Alexandra asked me… son, you know you’re a Finch, don’t you?”

“That’s what I’ve been told.” Jem looked out of the corners of his eyes. His voice rose uncontrollably, “Atticus, what’s the matter?” (Ch. 14)

To Alexandra, everything is about class. As a Finch, she is better than others in Maycomb.  She has a responsibility in the social order.  Scout and Jem also have it.  This is why she wants them to understand their position in Maycomb.