Times Square is a big open-air shopping mall/theme park.
I think what you mean to say is, Disney helped create a new marketing demographic that included families with children. Businesses saw the $$$ and followed suit (as well as a new breed of scam artists). But Times Square has ALWAYS been one of the biggest tourist destinations for shopping, dining and entertainment. It has always attracted huge crowds who stay in the area, hence the high concentration of hotels within Times Square. Since the Golden Age, there has been a massive increase in international tourism on a global scale as travel became more affordable, economies and communication changed and grew, and Times Square continues to evolve in an effort to keep up with the ever-increasing number of tourists. The new pedestrian zones evidence this. As soon as the new construction is finished (whenever that may be), we'll find that it will also not be enough to contain the crowds and will have to be expanded further.
But to the point, I don't really think it was "Disney-fication" that is responsible for casual attire in theatres (I remember seeing very casual attire pre-Disney). It's just the evolution of Times Square, tourism and societal norms.
Audiences at touring shows aren't subject to any of that, which is why they tend to dress more formally.
Perhaps that is the case where you live. In Chicago, you see the same gamut of attire that you do on Broadway at major houses, regional theatres and storefronts alike. I might be wearing shorts to Pippin tomorrow night, depending on what I'm doing at the time.
"What can you expect from a bunch of seitan worshippers?" - Reginald Tresilian