For smaller cuts and scrapes Neosporin is fine to use on your dog. The antibiotics that help heal minor abrasions works for dogs just as it does or humans.
Just like with humans, if the injury needs more than a couple of days to heal you should see a vet or doctor.
The only real problem with using Neosporin is that your dog will lick it off prematurely. The traditional treatment would include using an Elizabethan collar (aka “The Collar of Shame”) to prevent your dog from licking away the Neosporin. If ingested, Neosporin can cause an upset stomach, vomiting, and/or diarrhea.
There are more canine-friendly topical ointments that can be used instead of Neosporin, such as Vetericyn, but another option is using an alternative to the cone. Reusing old clothing items (sleeves from T-shirts, socks with toes cut out) positioned so that they cannot be pulled off can cover the Neosporin and prevent licking.