Opposition to the one is an article of faith for libertarians and economic conservatives (or, properly speaking, neoliberals). Opposition to the other is surprisingly muted, even though a hike in the minimum wage directly affects only some employers, while an increase in the payroll tax directly affects almost all. In each case, the outcome is to increase the price of labor, something predicted to lead to higher unemployment.
If anything, I would expect the remote effects of an increase in the minimum wage to offset increases in unemployment to a greater extent than the remote effects of raising payroll taxes would do. The minimum wage puts money straight into the pockets of people who will spend it quickly—and quite probably spend it in retail establishments that may be paying the minimum wage to begin with. The payroll tax, by contrast, takes money straight out of the consumer economy. The minimum wage helps (some) younger and poorer workers; the Social Security benefits financed by the payroll tax go to the old and relatively affluent. So on compassionate grounds, too, the minimum wage seems less objectionable than higher payroll taxes.
One might well oppose both, of course, but congressional Republican have tended to favor raising payroll taxes—they insisted on them as part of the Fiscal Cliff deal—and don’t come in for much criticism from the right for doing so, while the minimum wage is pure plutonium as far as the right is concerned. (So much so, in fact, that when Ron Unz makes a plausible case that raising the minimum wage will reduce immigration, even many activists who prioritize immigration over economic issues balk.) There’s criticism on the right of payroll taxes, certainly, but it rarely has the urgency of the attacks on minimum-wage hikes.
For the GOP there may be some short-term political logic in all this: the party wants to continue to offer older Americans—the ones most likely to vote Republican—as many benefits as possible, and plenty of influential GOP supporters aren’t paying any payroll taxes at all because they’re not salaried employees. But it’s hardly going to help the party in the long term if it keeps reinforcing the impression that it favors the rich over the poor and middle class—and even consistent neoliberals who dislike both payroll taxes and the minimum wage may think twice about such selectively small-government priorities.