No. Most minimum wage employers are service-sector businesses that are tied to a state because that is where their customers are—businesses like fast food, retail stores, and home health care services, for example.
Indeed, a New York Times profile of businesses on either side of the Washington/Idaho border (at a time when Washington’s minimum wage was nearly $3 higher than Idaho’s) found that businesses in Washington were flourishing, despite predictions that the higher minimum wage would send jobs and businesses fleeing across the state line. A similar recounting can be found in a report on Bloomberg.
In addition, a 2010 study by economists at the University of Massachusetts, University of North Carolina, and University of California provides further evidence. The study compared employment data among every pair of neighboring U.S. counties that straddle a state border and had differing minimum wage levels at any time between 1990 and 2006, and found that minimum wage increases did not cost jobs.