A production of Shakespeare’s “Julius Caesar” in Central Park featuring a look-alike of President Trump as Caesar. The host of an NBC News Sunday night show interviewing a conspiracy theorist who has claimed the Sandy Hook school shooting was a hoax.

Long gone are the days when an advertiser’s biggest concern was making sure it did not appear next to pornography online or during the ad breaks on racy TV shows. In this divided postelection world, brands are weighing in on theater and news interviews, as they grapple with the rapidly spinning wheels of social media, increasingly polarized consumer groups and a new set of potential ills beyond the usual list of sex, violence and crude language.

Delta Air Lines and Bank of America drew headlines this week for pulling their support from New York’s Public Theater in response to criticism about its production of “Julius Caesar,” in which the titular character — made up to look like Mr. Trump — is assassinated. Then, on Monday, JPMorgan Chase temporarily halted its ads on NBC News because of Megyn Kelly’s coming interview with Alex Jones, who operates the far-right site Infowars and has become more prominent because of his relationship with Mr. Trump. In both cases, the advertisers’ decisions were cheered by some and deplored as censorship by others.

“A lot of sponsorships that wouldn’t have garnered a lot of attention a year ago are now coming under greater scrutiny because people are wondering what that says about a business’s political stance,” said Kara Alaimo, who teaches public relations at Hofstra University. “Brands are going to be asking a lot more questions moving forward about the content of theatrical productions and potentially even of news outlets, which is sort of the more frightening prospect to me.”