Every time I walk from house to work and vice-versa I don't mind petting a couple of stray dogs I happen to bump into every now and then. I know they're stray cause there's no collars, and -sadly- I've seen them digging through the trash to find something to eat, I toss a snack every now and then. What I want to know is how risky is to adopt a stray dog and if it happens, what are the measurements(vaccines obviously, but besides that) I will need to take in order to successfully transform this stray dog into a home dog.
If you live an area with a humane society, dog pound, shelter, or whatever it may be called take the dog to them, express your interest in adopting the dog when you take it in.
They are experienced in matters related to stray dogs, they can scan for micro chips and handle any lost dog issues, as well doing a vet check and providing shots as need. Due to their nature I would expect the cost to be significantly less then the same services at your local vet. The dog may need to stay for 1 to 3 days while they do all of the things that need to be done.
Depending on the facility rules you may be able to visit the dog daily while it is going through the process.
Risk is always relative and no two dogs are alike. Determining the risk a dog carries isn't something you can assess clearly by yourself, but taking precautions against diseases an animal might have is a good idea (washing hands, gloves, etc).
Vet - Going to the vet is the first stop to check for diseases and likely worms. The vet can also check for microchips indicating the dog once belonged to someone. The vet also has good resources to refer you to trainers or classes if there are behavioral issues.
Other Pets, Children - Because you're adopting this dog off the street, there's no one to tell you if it will get along with other pets or any children.
Behavior - For your initial assessment of whether the dog is a good match for you and if you decide to pick it up, you're really on your own. The vet and professional trainers can help you diagnosis issues and help you decide if the dog is a good fit for you. Remember that a dog off the street is used to being able to come and go as he pleases, so they're going to have a lot more energy than your average couch potato dog. It's also worth noting that you have no idea how the dog got on the street in the first place - perhaps they were dumped there, perhaps they were abandoned for eating the previous owner's shoes, or perhaps the dog is a run-away (and therefore might run away again).