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.
Dove Care Sheet: Food, Habitat & Health
In case of emergency
Rescue - Wild Bird Fund
How to Tame Pigeons to Be out of Cage Permanently (with Pictures)
Dove for adoption - Pigeon, a Pigeon & Dove Mix in Indian Trail, NC
How to Catch a Pigeon or Dove in Need of Rescue
In case of emergency
Great Lakes Pigeon Rescue – Pigeon Rescue
How to Catch a Pigeon or Dove in Need of Rescue