The Congressional Budget Office has warned that failing to keep the tax increases and spending cuts from happening might pitch the U.S. economy into a recession. That would put pressure on state governments if sales taxes decline and residents lose their jobs.
"If we go into another recession, that will have a serious effect on everybody's revenue situation at the state level," Dayton said in an interview.
"We all depend on this economy expanding and creating new jobs and generating revenues and lessening some of the burden on some of the social programs," Dayton said. "We all stand to gain from that, and we all stand to lose if it doesn't happen."
States would also be affected by automatic budget cuts set to begin if Congress doesn't come up with other ways to reduce the deficit. They would lose $7.5 billion next year from dozens of programs that flow through states, including money for public schools, according to Federal Funds Information for States, a Washington-based group that follows the budget for states.
That represents a small share of the approximately $575 billion they received in federal funds in 2011, according to the National Association of State Budget Officers.
Utah's Herbert said it's important that states receive more flexibility to manage state programs if the funding is cut.
"States are willing to do our part - we understand this has to be a shared sacrifice," he told reporters. "States are willing to do more with less."