Chris Edwards says “yes.” From Downsizing the Federal Government:
Another myth is that policymakers cannot make budget cuts without a backlash from voters. Yet reform efforts in the 1990s did not lead to a voter rebuke. In 1996, the Republicans were denounced viciously when they were reforming welfare. But they stuck together and succeeded, and today the achievement is widely hailed. Also in the 1990s, the Republicans proposed reductions to many sensitive programs including Medicare, Medicaid, education, housing, and farm subsidies. In their budget plan for 1996, House Republicans voted to abolish more than 200 programs including whole departments and agencies.
The Republicans who led on these reforms were not thrown out of office, despite many of them being specifically targeted for defeat in 1996. The most hardcore budget cutters in the 104th Congress were freshmen who were reelected with larger vote margins than they had received in 1994. They included John Shadegg and Matt Salmon of Arizona, Joe Scarborough of Florida, David McIntosh and Mark Souder of Indiana, Steve Largent and Tom Coburn of Oklahoma, Mark Sanford of South Carolina, Van Hilleary of Tennessee, and Mark Neumann of Wisconsin. Indeed, many budget-cutting Republican freshman got reelected in districts that went for Bill Clinton on the presidential ticket in 1996. The high-profile leader of the House budget cutters, John Kasich (R-Ohio), consistently won reelection throughout the 1990s with two-to-one margins. In sum, cutting the budget can be good politics when done in a serious and up-front manner.