They're not suing them in civil court, they're threatening to use the court of public perception against them. If they allow these payments the activism groups will set up a campaign titled something like "Visa facilitates incest and child abuse!" and "Mastercard allows you to see women getting beaten".
This is a very effective strategy because there's nothing more important to these companies than a squeaky clean brand image. And what they perceive as damaging to their brand image is entirely subjective and just depends on whether an activist group can spin it in a way that looks bad for them.
