An increase of the minimum wages in the United States would render some of the manufacturing non-competitive as compared to their foreign rivals (who may be able to produce it cheaper). It would make sense for many companies to simply outsource (or send abroad) some of these minimum wage jobs to countries where wages are less. An increase in minimum wages may also prompt many businesses to hire illegal immigrants, who may be willing to work at a rate lower than the minimum wage.
Simply put, increasing the wages by a certain fraction does not increase the labor productivity by similar percentage, i.e. if the minimum wages are hiked by 20%, the company's production will not increase by a commensurate measure. This causes cost-cutting at other fronts, in terms of job-loss, outsourcing or illegal immigration (undocumented immigration).
An increase in the minimum wage will mean that firms will likely have lower labor demand, while the labor supply is high (more people would be willing to work at higher wages). In terms of elasticity of labor demand:
wage elasticity of demand = % change in employment/% change in wages
An increase in wage would reduce the employment, unless the demand is unit elastic (=1). The level of elasticity will determine how much impact the change in wages will have on employment level.