This has come up recently in conversation and I think it’s an important topic. First, I would say avoid approaching a stray if at all possible. Call animal control and try to gently confine the dog but don’t touch. But where the dog is in imminent danger or is hurt, it may be the right choice to try to catch him. I think anyone is entirely correct to be cautious. You don’t have to be ballsy or foolish to be a good person with dogs.
If you think you may someday have to catch a stray, you should make up a stray-dog kit and keep it in your car, and maybe tuck another one in your purse. Put some shelf-safe but smelly food (in my experience dogs go nuts for beef jerky) in a little bag and get and roll up a light kennel lead. A kennel lead is that type of lead that vets and animal control officers use–it’s a very light poly or webbing lead with a ring on the end. You can bother to get a commercial one or you can get any 6′ length of light cording and tie a loop in one end, then stick the other end through the loop to make a noose.
You can attract the dog with the food, and nice happy soothing voice, and then gently slip the noose over the dog’s head; you can sort of hold it open in a big loop and just drop it over the head and pull up on the slack to close it on the dog’s neck. Once leashed, most dogs are pretty much under control. Then call animal control or the police–I’d say that you shouldn’t pick up the dog even if she seems friendly. Dogs that are gleeful and affectionate on the ground often have hot buttons like tails or feet, and you’re a lot more likely to stress the dog if you pick it up. Just keep the dog safe and get it to the proper people.