Where I grew up it was a common question to ask when seeing a friend soaking wet was, "Did your Mom ever teach you to get out of the rain?" Here in the city there seems to be a real void on this saying.
People here will stand in the rain waiting for anything and everything! Where I get off the train there is a coffee truck that will have 5 people in line no matter how bad the deluge. It is not only coffee people will get wet for, it is burgers, shakes, cupcakes, pizza, sample sales, designer boutiques sales. and more.
I always thought it was only the tourists that would stand in long lines but I was wrong. The main difference is that tourists wait to see the sights like The Empire State Building and New Yorkers wait to eat or shop. I even heard about one woman who was walking down the sidewalk and saw a really long line so she just got in the line. She didn't even know what the line was for but told someone if the line is this long, it must be good.
You might think that these people have an umbrella. It is much more typical to see people standing in the rain in their waterproof jackets without an umbrella. Umbrellas can be helpful but often just get in the way, especially in line since they tend to poke your neighbor. If you fill the need to get an umbrella buy the cheapest thing you can get on the street. All of the street vendor umbrellas are pieces of junk so don't feel like your investment will pay if you get a bigger one. One little tip, you can get a decent umbrella (as long as it's not too windy) at just about any 99 cent store.
When you are here you will rely heavily on your feet and so you can't allow the rain to ruin your day. You will be outside a lot and have to be ready for the rain. Wear rain boots or something water resistant if you don't like we feet. No matter what, suck it up and get out there and get in line. There is too much to see, eat and buy to stay home in the rain or snow for that matter.