One thing that it would curb even in social structure is philosophical differences. One school maintaining that souls are imprisoned in matter as punishment -- one school holding that souls are created with the bodies that they animate -- one school holding to reincarnation as a punishment or reward according to situation you are born into, and another that it is a way to give every soul a chance at every situation, so that no soul, in the end, will be able to complain of not being given the same chances as other souls. If you have the planes, all the other schools are just being stupid.
It doesn't work in many ways in society as it stands now. It would be a big change.
Of course, there are other changes. I was thinking of the effects of "detect evil" and various anti-falsehood spells. Those would change society a lot. Governments could be a lot less corrupt with liberal usage -- to be sure, tyrannies would also lack as many cracks to slip through.
It would make D&D thieves' guilds less tenable. OTOH, it would make the life of adventuring thieves easier. Everyone knows that they have to be honest in civilized lands.
Much depends on how heavily the magic is used, of course. But D&D is obviously a high magic area, because infant mortality is not affecting all of society.