I always thought this was a class issue. Plus, the sign has a certain touching alliteral poeticism:
No Shirt
No Shoes
No Service
See, all the first words are the same, and all the second words have the same first letter. 
Very witty, no?
It just appeals to the owner as right for his establishment. It’s now almost a tradition for cheap restaraunts, isn’t it?
I also suspect that this sign is readily available and prominently displayed in the store fixtures catalogs and wholesale stores.
(hijack) One sees the same thing with the ‘For Sale’ sign. It’s readily available, and seems to be the right product. In reality, the best way to sell your car by the side of the road is to put a price in the window with big numerals. Put your phone number in smaller numbers, too. If the car is on the side of the road with a price in the window, it’s obvious it’s for sale.(/hijack)