Raising minimum wage shows some promise at improving things, but it seems rents for in-demand cities simply increase to match the median income of workers in the area.
Re-shaping tax laws has a much better chance of improving matters for low-income workers. If capital gains wasn't so low, running business would again become better investments than stock markets. If dividend stocks were tax-advantaged and became more common, then stock-holders would have a much greater interest in long-term stability of a company rather than short-term returns. If the taxes on property weren't so low, the ultra-wealthy wouldn't be buying up every cheap bit of it and holding onto it for decades, perhaps renting it out rather than ever selling it. If there weren't so many tax law loopholes, huge corporations wouldn't have as much of an economic advantage over smaller businesses.
I don't subscribe to the notion that impoverished workers are required by our economy, either. While the Wal-marts and Sam's Clubs keep people barely surviving, the Targets and Costcos out there give their workers a reasonable standard of living and don't need to sell lots of expensive luxury goods to pay for it.