I was digging up guidelines recently and saw that some set of criteria are required for these. Which ones?
The partner program is currently obsolete (they ended it last year), the badge is just a memento for people to keep for participating at the time. It's possible they bring it back in the future, but no plans currently.
For becoming a mod, the existing notes in the site guidelines aren't really relevant anymore. Basically, if you have been active on the site for a long time & are considered to be a reliable, positive-contributing member of the community, David is often open to adding new mods, so you can always ask him. But in practice it's more often something you're asked to do than vice versa (mod accounts are given powers that could mess with the site, so usually he's only looking for long time known/trusted users. And usually it's him asking people because being a mod doesn't give any consistent benefits, there's no real reason to be one unless you have been using Manifold for a long time and want to help maintain the site).