About a tenth of southern whites, about a third of them, and about all of them are all correct answers because this is more complicated than it appears at first glance.

By the most restrictive definition, and using the best all-South survey we have, we can say ~10% with considerable local variation. Enslavers are much more common in the Lower South (the cotton states) than the Upper South (hemp, tobacco, wheat, etc). You can pull that right out of the 1860 US census and call it good, but that's the worst answer of the three for a bunch of interrelated reasons.

The census only lists heads of households as enslavers. Those people mostly deserve to be listed, though some overseers who work for absentee enslavers and don't own people themselves probably get counted too. By the strictest definition, an overseer is not an enslaver. (This is problematic for reasons forthcoming.) The head of household definition of enslaver obscures the fact that the single person who holds legal title to an enslaved person is not the sole beneficiary of their stolen labor and life. They usually have a spouse, children, relatives who live with them, etc who all benefit from slavery in a direct, day to day way. The enslaved are people, but they are also property and available to the whole family, essentially. Cars and houses aren't people, but the same issue applies here: growing up you might not own a car or a house but your parents might and you live in the house and get the use of the car. And that's before you get into practices like the whole family having direct supervisory power over enslaved people.

It might legally be true that the head of household owns the people, but they're not the only one exercising the prerogatives of ownership or realizing profits from it. So we can get a better look at the scale of slaveholding by checking into the number of enslaving households. Back in our 1860 census we can take the number of slaveholders and divide by the number of white households to get a ballpark number. (We will miss the small number of black enslavers in doing this, but they're not even one percent in any southern state.) That brings us up to around a third, still with fewer in the Upper South and more in the Lower. In South Carolina, an outright majority of white households enslave people.

The one third number is basically the standard answer because it gets at most of what people mean when they ask how many southerners owned people. They want to know how often, if they walked into a random Southern house, they would find a slave in normal residence. It's a pretty good measure for that, but already from considering the census definition of slaveholder we know that the boundaries are fuzzier than the numbers suggest. They get fuzzier still.

Take it as given that most to all whites in an enslaving household have some degree of control over the people they enslave. This can be formal, as in the head of household giving an enslaved person to another's charge, tasking someone to oversee the whole operation, putting someone else in charge of the kitchen, etc. With fewer enslaved people, things become less formal but the same basic principle applies. The whole house is participant in enslaving and might exercise the prerogatives of ownership, up to and including delivering punishments (or getting others to do so) and sexual violence.

The thing is that a lot of that is actually true for most every white person around in a broader sense. The slave codes -laws which regulate slavery- and general custom endow white people, mostly white men but also whites in general, with vast power to challenge slaves who are abroad. They can be called upon to aid in slave patrols, where they look for runaways and can apprehend anyone who doesn't answer to their satisfaction. Those people can legally be whipped, beaten, etc by the patrol in the course of their duties. Whites are also largely immune to punishment for offenses against slaves especially and black Americans in general through the legal disabilities placed upon them. An enslaved person can't give testimony against a white in court, for example. The law usually presumes any visibly black person to be enslaved until proven otherwise on top of that. This has the intended effect of removing black Americans of slave or free statue from the protection of the law, at least insofar as it applies to offenses against them by whites.

The full package of benefits given to white skin can be nebulous and hard to imagine, but it's massive and of tremendous value to the people who possess it. The simple fact that however modest one's means -including miserably poor and desperate- the slaves occupy a still lower run to which you can never fall really means something. There's a pride in status there, and a kind of equality for white men built on the deprivation of the enslaved. It's not prefect or as idyllic as proslavery theorists let on, but it really does matter and people do rationally seek to preserve those advantage even to their sometimes serious detriment.

Considering that slavery is the central, civilization-defining institution of the antebellum South it's very hard to find people who don't fall under the broadest count. It's literally a slave society and slavery touches most every life deeply. The chief exceptions, and they are only partial, come in areas not suited to plantation agriculture or to which it has yet to spread. There you occasionally find slavery-adverse whites who are willing to buck the system, yet even then it's almost entirely as a tactic to extract concessions from the richer plantation belts. These are often in the mode of "give our region a bit more say in the state legislature or we'll go antislavery on you." In that sense, even they are exerting a kind of enslaving power. As soon as they get what they want, or at least some of it, the challenge goes away.

On the balance, the one third number is the best answer but it's important to also keep in mind that you have to look really hard to find someone in the slave states who isn't materially and psychologically touched by slavery whether they literally own someone, live in an enslaving household, or are just present while white. Slavery is just that ubiquitous. If you want to get really broad, you could even point to free laboring farmers in the Midwest, bankers in the Northeast, and the operators of textile mills in New England, England, and France and all their consumers as part of the same vast system for turning torture into money. One needs to note the many nuances and complexities in doing that, but the basic idea is sound and the point was made by people at the time.