The phrase "run down boarding houses" can be replaced with a variety of synonyms, depending on the context and tone of the writing. One possible option is "ramshackle hostels," which connotes a sense of disrepair and dilapidation. Another choice could be "seedy lodgings," which suggests an unsavory and perhaps even dangerous living situation. "Dilapidated rooming houses" is another possibility, emphasizing the worn-down and abandoned nature of the accommodations. Ultimately, the synonym chosen will depend on the desired effect of the writing, whether it be a harsh critique or simply descriptive language.