You can catch a pigeon that needs help in a way that's safe for both you and the bird. Here are a few methods with a good success rate.
Palomacy Help Group for Pigeon & Dove Rescue & Adoption
Dove Releases - Dreaming of a Chance Bird Sanctuary
Pigeons – Dirty Birds or Wartime Heroes?
Palomacy Pigeon & Dove Adoptions: ***SELLING OUT FAST!*** 2021 Palomacy Calendars: Wall & BirdADay
How to Help an Injured Wild Bird
Dove for adoption - Penny, a Pigeon in Chicago, IL
Wingclaps for a Winning Pigeon Pocket Guide - American Birding Association
Found a Sick or Injured Pigeon? Here's What to Do (and Not to Do) - PetHelpful
How to Take Care of a Lost Pigeon: 10 Steps (with Pictures)
Pet Dove Care Sheet & Supplies
I handled a wild pigeon without gloves (UK) : r/pigeon
Rescue - Wild Bird Fund
In case of emergency