'Brought low' typically means 'to come to experience meaner circumstances from a background of economic privilege, influence, power, etc.; humbled'. And, mind you, not 'humbled' as in 'gratified and appreciative', but more 'crushed underfoot', as per Biblical usage:
And the loftiness of man shall be bowed down, and the haughtiness of men shall be brought low; and Jehovah alone shall be exalted in that day
Anyway, I agree more or less with aeismail. In this context, 'brought low' does mean 'having been in some bad situations', but the subtext is missed. The connotation is that she was a 'good girl', one from a relatively well-off family, who had come to face some circumstances that a girl of her standing normally shouldn't. Consequently, she might have done some things or had some experiences that be cause for shame, considering societal expectations of her, and the expectations her former life would have led her to have of herself.
As for a word to replace brought low, 'humbled' could do it. A bigger bit of re-phrasing could be done, too:
...if the girl had a secret, as nearly all girls have, having crossed the tracks as she evidently has-it had nothing...