Hays has weathered the uncertainty better than many places. Generations here have enjoyed band concerts and theater on the campus of Fort Hays State University, and the Arts Council claims to be the oldest arts-supporting organization of its type in Kansas. Both the city and county governments provide funding for the council, which has an annual budget of roughly $125,000, and residents of all political stripes cite the creative scene as a point of pride.

“A lot of my good Republican friends are out there, conservative and moderates, and they’re taking it in, they’re going to all of the venues and taking in the live music and the art,” said Henry Schwaller IV, a Democrat who serves on the Hays City Commission and who was chairman of the Kansas Arts Commission when Governor Brownback withdrew its funding. “It’s just part of our DNA here. And that’s hard to replicate in other communities.”

The board president of the Arts Council, Mike Morley, 47, is a registered Republican who voted for Mr. Trump in November. Ms. Meder, 60, is a Democrat who voted for Hillary Clinton.

“On the political spectrum, we’re right to left and everything in between,” said Mr. Morley, who moved to Hays about five years ago and works in the utility industry. “But the one thing that we all have in common is an appreciation for the importance of the arts.”

Here in Ellis County, nearly 71 percent of voters favored Mr. Trump, part of his sweep of 103 of Kansas’s 105 counties. In interviews, some of those voters cited the president’s commitment to deregulation, their frustration with traditional politicians and concerns about government spending. But when it came to arts financing, the conversations were more nuanced among Trump voters. Some said they supported the proposed cuts, but others questioned whether eviscerating the endowment was the right approach.

Jacob Brubaker, a sophomore at Fort Hays State and a member of the College Republicans, said that the federal government should perhaps stop funding the arts eventually but that the drawdown should be gradual so that artists could find other money.