I've seen a few articles now discussing the fact Mike Pence won't eat dinner alone with women other than his wife, the best being in the Atlantic:
https://www.theatlantic.com/science/archive/2017/03/pences-gender-segregated-dinners/521286/; https://www.theatlantic.com/politics/archive/2017/03/pence-wife-billy-graham-rule/521298/
Evangelical preacher Billy Graham used to have the same personal policy - he would never be alone with any woman except his wife. Graham writing in his autobiography that he was worried about quote unquote, the "the temptations of sexual immorality." That's
clearly problematic - it's perpetuating the belief that being around women is inherently a temptation to men, and the notion that men can't help their actions when it comes to intimate relations.
It's also perpetuating the false notion that men and women can't simply just be friends and enjoy spending time together without it being inherently romantic. That false understanding can become particularly toxic when applied to public figures -
For example here in Australia, claims by a journalist that the former Australian Prime Minister Tony Abbott had been seen at a restaurant with his Chief of Staff Peta Credlin where he'd eaten something off her fork led to allegations that the two were having an affair and some, I'll say interesting, comments on Social media. I can see someone in Mike Pence's position wanting to avoid that sort of media discussion. A hard line Evangelical governor with traditionalist and certainly homophobic views about marriage and family caught having an affair would give the papers a field day.
But still, as the two Atlantic articles point out, the notion is inherently sexist and brings about negative consequences. If he doesn't want to eat alone with women, how is Pence, the Vice President, supposed to meet with female Legislators, staffers, lobbyists ect, particularly in an environment like Washington where a lot of deal making is done in those informal settings. President's have been known for instance to give rides in Air Force One to Members of Congress to get them onboard with legislation.