Undoubtedly, the cost of housing has been rising relative to incomes, especially for young and poor workers. There was a slight reprieve when the housing bubble collapsed, yet the unaffordability continued to climb quickly thereafter.
Reduce income inequality, build smaller homes that can be owned instead of apartments that can be rented, and improve public transportation.
First, it's important to understand that the high cost of living isn't really a problem. It's a good thing. I mean your city, town, or neighborhood is a desirable place to live, and a lot of people want to move there. The real problem is that as you say you have a lot of young and poor workers who struggle with it. A major city does need unskilled labor to work in the restaurants and shops that the wealthy urbanites visit for lunch and happy hour. You either need to pay them well enough that they can afford to live downtown, subsidize their living, or provide cheap affordable transportation so that they can easily get downtown while living in a cheaper suburb that's farther away.
Second, one of the big problems particularly for younger single professionals is that so many of the homes that are built are designed for families. They're huge 3-5 bedroom units that a single young professional can't afford unless they rent out the other bedrooms to friends. Meanwhile, right assholes with too much money laying around buy up a ton of properties for rent not to mention build all kinds of big apartment complexes. As a young professional that wants to live alone, you get stuck renting a 1 bedroom apartment forever because you can't afford a down payment on the houses that are too big for you anyway. Even though your rent payment is almost the same as a mortgage for a decent sized house, you have to keep throwing your money away on rent because you can't save for the down payment, and without a down payment, the mortgage insurance you'd have to pay makes any home unaffordable.
Imagine instead of all these giant 3-5 bedroom houses that are going for over $400,000 or an apartment complex that only allows you to rent a 1 bedroom we saw smaller condos and townhouses that were 1-2 bedrooms max. They could sell, not rent, for $100,000-$200,000 and be well within the range for someone younger or poorer. They would need lower down payments, and could potentially pay the loans down faster and create equity. That way when they actually got ready to settle down and start a family they could sell their tiny homes and already have a good start on something bigger.
I'm a perfect example of what I'm talking about. When I first moved to Charleston, SC a few years ago I tried to buy a townhome. I didn't need a two bedroom, but those were the smallest that was available to buy. I could have easily afforded the mortgage on it, but I didn't have much for a down payment because between rent, student loans, and a car payment saving money was virtually impossible. As a result, I couldn't get the townhome without paying mortgage insurance which combined with the HOA fees put it well out of reach.
The result is that three years later I'm still renting. Even though my rent is damn close to what my mortgage would have been, and even if I get married it's unlikely I'll be able to buy anything for at least another two years despite making close to 6 figures.