WASHINGTON—House Speaker John Boehner said Thursday that he's determined to offer a budget this spring that curbs Social Security and Medicare, despite the political risks, and that Republicans will try to persuade voters that sacrifices are needed.

In an interview with The Wall Street Journal, Mr. Boehner said House Republicans would offer a budget for the next fiscal year that sets goals for bringing the programs' costs under control. But he acknowledged that Americans aren't yet ready to embrace far-reaching changes to...