Having had a similar discussion with a bunch of people who are not fans of dogs, I think the key to responsible dog ownership is respect.
Respect your dog - treat it fairly, consistently and appropriately. Provide suitable outlets for exercise of body and mind, and ensure your dog is healthy and well fed. Ensure you have trained your dog to a sufficient level to allow it off-lead exercise as appropriate to the breed. Ensure you know enough about the breed to recognise any breed specific needs/issues and deal with them appropriately. Learn how a dog communicates and how to motivate it to respond to you. Don't leave your dog tied up outside shops/schools etc. Don't leave your dog alone for excessive periods of time.
Respect other dogs - don't allow your dog to approach on-lead dogs unless invited to do so by owner. Don't allow play to get out of hand into bullying, don't allow aggressive dogs to attack other dogs. Don't take your in-season bitch out to popular walking spots in the middle of the day.
(While I understand Ninasmum's view that all aggressive dogs should be muzzled/on-lead, it's a difficult one, as it depends on the dog and the degree of aggression... not all owners know the difference between a grumpy dog, a reactive dog and an aggressive dog... I know some owners of dogs who've been on the receiving end of Honey on a bad day have said we should have had her on-lead and muzzled, yet you know that generally speaking she isn't actually aggressive - and to have reacted initially by keeping her on-lead and muzzled would not have solved her problems... but I guess that comes down to understanding more about dog communication and body language and realising that not all dog disagreements are because a dog is aggressive... very difficult!!!)
Respect other animals - don't allow your dog to worry livestock or other animals. Ensure your dog is either under close control or on-lead.
Respect other people - be aware that not everyone likes dogs. Don't allow you dog to jump up at strangers, run towards people (especially children), lick babies in buggies etc. Don't allow your dog to raid picnics, or to chase cyclists/joggers etc. Always pick up poo (or flick it into undergrowth where this is more appropriate) and dispose of it properly. Keep your dog on a short lead on all roads, even if your dog walks to heel perfectly - drivers don't know your dog!!
Respect yourself - get a dog that is appropriate to your circumstances, and ensure it is well bred and health tested (if a puppy) or comes from a reputable rescue. Learn about your dog and enjoy its company. That way you are more likely to succeed at raising a well-adjusted, well-trained dog!
I'm sure I've missed out some specifics, but that it the gist of my dog owners code.
It would, however, come alongside a general dog code, where the public are also asked to respect dogs, and to teach their children to respect dogs, so that children know not to run around screaming when they see a dog, and that people know not to approach dogs unless the owner says it is ok. (And especially not to say "Oh, it's ok, I don't mind him jumping up" or "Here, have a biscuit"
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