Try and develop a consistent system of warnings - and then use the auto-ban function when someone gets too many warning points.
My approach is to give someone an "educational warning" for their first offence on a particular matter where I explain the problem, point to the section of the rules which covers the matter and explain what action has been taken (eg post edited / deleted / etc).
For subsequent offences on the same matter, they get warning points assigned and a more harsh message cautioning them that continued behaviour of this nature will see them banned from the site. Warning points expire after 12 months.
If they receive sufficient warning points, the system will either temporarily ban them for 1 month, or permanently ban them.
I've set it up so that three serious infractions (after their initial educational warning), will see them permanently banned. For extremely serious matters, I may bypass this system and directly ban someone. For minor infractions, they receive less points - so it may take longer to ban them, but if they continue breaking multiple rules - it all adds up.
I like this system because it removes most of the emotion from the process of banning someone. Basically, you let the system decide when they get banned - all you do is hand out warnings (but it is important to try to be consistent in how you apply those).
Then you can tell other members that the system banned them for receiving too many warnings.
I also have a policy to not discuss banned members in public - it's especially important on one of my sites because people are generally not anonymous and so some comments that come from these types of discussions can be defamatory and lead to legal issues. I'm happy to answer questions directly and factually about someone in private (without going into too many details), but not in public discussion.
That's my approach anyway.
I've been running forums for 20 years now - I've found the best approach is to try and be as consistent as possible. I'm also pretty strict - but I try to be fair and consistent, knowing that you can't keep everyone happy all the time. This is also why I don't let my moderators hand out warnings - I make the decisions on who gets a formal warning, so that personal biases (other than my own!!) don't enter into it.
Moderation is a tough gig - you're always criticised for either moderating too much, or not moderating enough - quite often at the same time!