The Sawbones badge is for "Reviving a friend 50 times"
When a co-op partner gets downed (reaches 0 health) they will have a blood icon above their head. At this point you can go over and interact them to revive them.
The optimal way to do this is; get to the end of the "grind the bad apples trial" and repeatedly walk into the grinder to die, then have your teammate revive you (this will also help them progress towards the "Get Down" badge). Repeat this process then swap with your Co-op partner.
Also two handy things to note:
1. The in-game badge menu tracks your progress.
2. Leaving a trial after gaining progress still counts towards it.