I see arrogance as actions where a person does not have or show empathy/understanding towards others. So, in general, people should not try to be less arrogant (that would be a fools errand). Instead people should focus on developing empathy.
So find times throughout the day to pause and reflect on what you've said and done. Ask yourself how others might interpret what you've said and done and if that could have been done in a different way that still accomplishes the same thing while being more understanding of their situations/circumstances.
Once you have developed that habit you can start to strategically craft that questioning into everything you do/say proactively.
Keep doing this continually and ideally you will develop more empathy.
Edit: also find time to regularly volunteer in some way. i.e. if all of the things you do are towards one end - helping yourself, it's difficult to believe you're actually an empathic/caring person. Sometimes you have to live it to become it. So do it and then you'll be it.
IMHO.