On 16 October 2017, the New Yorker ran a feature (“The Danger of President Pence”) that described what a potential Mike Pence administration would be like should President Trump leave office before his term is up. As an illustration of differences in their views on social issues, the piece recounted a meeting between Trump, Pence, and a legal scholar in which Trump allegedly “joked” that Pence wants to “hang” homosexuals:
Two sources also recalled Trump needling Pence about his views on abortion and homosexuality. During a meeting with a legal scholar, Trump belittled Pence’s determination to overturn Roe v. Wade ... When the conversation turned to gay rights, Trump motioned toward Pence and joked, “Don’t ask that guy — he wants to hang them all!”
The White House provided a statement to Politico denying that the President made such a joke, with Press Secretary Sarah Huckabee Sanders also averring that “the suggestion that he would make such outrageous remarks is offensive and untrue.” A press representative for Vice President Mike Pence told the Indianapolis Star that:
Articles like this are why the American people have lost so much faith in the press. The New Yorker piece is filled with unsubstantiated, unsourced claims that are untrue and offensive.
We reached out (without response) to the spokesperson who made this comment to clarify whether this statement was a specific denial of the claim that Trump's "joke" was made, rather than a broad critique of the piece as a whole. For their part, the New Yorker said they stand unequivocally by their reporting of this alleged joke, telling Politico:
In the course of fact-checking this piece, we talked to more than sixty people to confirm the reporting contained therein, including senior White House officials, a senior member of the Vice-President’s office, the RGA, Rep. Elijah Cummings, and multiple people who were in the room when President Trump joked that Vice-President Pence ‘wants to hang’ gay people. We stand by the story.
Regardless of the specific incident discussed in the New Yorker piece, Vice President Pence has, in his actions and statements, shown himself to be hostile to gay rights during his career as a politician, as noted in a 2016 New York Times overview:
As governor of Indiana, Mr. Pence opposed gay marriage and signed into law a bill that made it legal for businesses to cite religious freedom when refusing service to gay and transgender people, for example a bakery that refused to bake a cake for a same-sex wedding.
As a member of Congress, Mr. Pence voted against employment nondiscrimination protections for gay and transgender people and also voted against the repeal of “don’t ask, don’t tell.”
His opposition to gay rights goes back at least to his first congressional campaign, in 2000, when he opposed same-sex marriage and the nondiscrimination laws that protected LGBT people.
He also argued for public funding of conversion therapy and said federal funding for HIV/AIDS treatment should be renewed only if the government could certify that no money went to “organizations that celebrate and encourage the types of behaviors that facilitate the spreading of the HIV virus,” according to an archived version of his campaign website.
Such views are presumably what would have led President Trump to make such an ill-considered (and widely condemned) “joke" making light of the persecution and execution of people based on their sexuality. However, as far as we know, no one (apart from possibly Trump) has ever alleged that Vice President Pence is actually in favor of hanging gay people.