ALBANY, N.Y.—Sen. Bernie Sanders is taking some of the boldest and potentially riskiest steps of his campaign ahead of the crucial New York primary, hoping to raise his profile and undercut Hillary Clinton’s support among voters who are a key part of the Democratic electorate.

After his debate with the Democratic presidential front-runner on Thursday, Mr. Sanders will leave the U.S. to appear at an academic conference at the Vatican devoted to themes of economic and social justice. Both issues are at the core of Mr. Sanders’s...