The government has the right to dictate how a business is run through the Interstate Commerce Clause, Article 1, Section 7 of the United States Constitution and furthermore through the Civil Rights Act of 1964 declaring that "'commerce' means travel, trade, traffic, commerce, transportation, or communication among the several States, or between the District of Columbia and any State, or between any foreign country or any territory or possession and any State or the District of Columbia, or between points in the same State but through any other State or the District of Columbia or a foreign country."
In essence, through these two documents, the Federal government established the right to govern all businesses since all businesses inherently are involved in some form of interstate commerce. This is why a store-owner cannot refuse service to someone based on ethnicity. Furthermore, through the 10th Amendment of the United States Constitution, "The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people."
In sum, there are hundreds of cases that have reached the Supreme Court involving government's role in private business and thousands on the District and State level. There is a tug-o-war that exists which can be best viewed in the Federal Monopoly laws or The New Deal (instances where a lack of government involvement lead to societal detriment). Many times, government has stepped-in for the public good and these measures were met with resounding approval.
Now, it is important to debate whether or not government has the right to determine if a privately owned business servicing the general public can allow smoking but to question the right of government to pass legislation regarding how a business is run is kinda silly. Of course they can.