What does "RUN EASY BOSTON." mean?

This is a good question because the sentence is somewhat ambiguous. I assume it means something like "Run, people of Boston, but use less than maximum effort." The problem is with the use of the word easy. The ad should say "Run easily Boston." or even "Run easily, Boston." because easily is an adverb. An adverb directly modifies a verb, such as run.
The word easy is usually an adjective which is a word that modifies nouns. If the advertiser really meant to use the word easy then "Run easy Boston." would mean something like "Take control of the part of Boston that is casual and laid back or of loose morals." The advertiser is a shoe company so this meaning is very unlikely.
The word easy can sometimes be used as an adverb or something very much like it. Take the phrase "take it easy" for example. Easy is clearly modifying the verb take. How odd would the old Glenn Frey/Jackson Browne song sound if it was called "Take it easily." The word easy is used as a noun in New Orleans when they refer to the city as "The Big Easy." So usage is as usage does. Still, advertisers should endeavor to use currently correct grammar because the public may adopt their usage.
I say currently correct grammar because correct grammar changes over time. Languages are living things that evolve in many ways for many reasons over time. Usage panels exist at companies that publish dictionaries and grammar books. Their decisions about correct usage will sometimes change because popular usage has irreversibly changed. For example, it used to be incorrect to use the word since when you really meant because. But, because so many people would use the word since for the second word in this sentence many usage panels now consider it acceptable. So people who are obsessive-compulsive about grammar are fighting a losing battle. Grammar will never reach a perfect pinnacle from which it will never change. The real value of usage panels is to slow down the change of grammar so that most speakers of a language can understand each other over the course of, say, a lifetime.