Dog Health Questions? Ask a Dog Vet for Answers ASAP
You can safely use peroxide, chlorhexiderm, or betadine to clean the base of the nail. Neosporin or other topical antibiotic ointment is ok to apply topically. Be sure to distract him with some play or a short walk or he will immediately lick the Neosporin off. It won't harm him in any way, but it takes a few minutes for absorption to offer any kind of protective effect so make sure you distract him long enough to give it time to work.
If he is chewing/ licking excessively at the toe, put an E-collar on him if you have one, or cover his foot with a sock (provided you're sure he won't eat the sock!) and tape it to his skin. Clean the wound twice a day and allow supervised time with the sock off so that it can dry out.
If you are not seeing improvement in a day or two, you should have the nail evaluated by your vet to make sure there is no infection or other serious problem present.