The relationship between ordinary Northern whites, Northern political organizations, other institutions, and politicians and slavery is complex and a full accounting of it is more than half the political history of the United States to 1860, but the claim you've heard is a narrower one.

Reformulating to isolate the individual aspects of the statement, have: The North during the Civil War Era was equally guilty of and responsible for slavery. So we're talking about the North from about the time of the Mexican War until 1860 or 1865. Let's make it an even 1840-1860 because after 1860 things change rapidly.

For the North to be guilty of slavery during this time, it needs to have some slaves. Sure enough, it does. All the way until 1860, slaves live in New Jersey. I don't mean just enslaved people passing through as enslavers use their rights of sojourn and transit, but actual people owned as property under the laws of the state. This is because of gradual emancipation, the process by which most of the North became free. The basic deal is that every person born to an enslaved mother after a fixed date must serve out a term of years to their mother's enslaver, but then become free. This usually means they're enslaved until their twenties. Everyone born before that date remains enslaved for life.

This means that most northern jurisdictions have some enslaved population (and sometimes tens of thousands) for decades after becoming "free" and then remnants of older slaves who were born too soon. Once the enslaved population shrinks enough, the state usually comes through with a second law that frees those who slipped through the cracks. New Jersey never did, so in 1860 it still has 18 slaves. At the same time the slave state with the lowest enslaved population, Delaware, has 1,798. That's just shy of a hundred times as many. Delaware is freakishly un-enslaved by the standards of the South, which averages about a third of its total population enslaved. If you want to compare apples to apples, and construe the North as broadly as possible by including all the nation's territories in it, then we can push the number of enslaved people in the section to 61. The entire South has an enslaved population of 3,953,696. Prior censuses have similar numbers, if with a bit more slaves in the North in absolute terms, which I can go into if you'd like.

For the North to be just as guilty of slavery as the South, we would expect that northerners would enslave about as many people. It doesn't have to be exactly the same, but one would hope for at least a general ballpark. This is not that.

Northern responsibility is a more complicated concept. White northerners have a long history of participation in institutions in partnership with the enslaving South from business ventures to churches to political parties and, of course, the United States. One could argue that Yankees profit from slavery indirectly through those partnerships. They also write loans that accept slaves as collateral, including loans to buy more slaves. Debt to northern financiers, especially after the Panic of 1837, further limits the freedom of southerners to set their own agricultural course since creditors want cotton planted for the best returns. Northern textile mills run on Southern cotton. Northerners produce extra-shoddy goods specifically, especially clothes, for enslavers to give to their slaves. The Midwest sells much of its farm products into the Lower South, where they have a market because Southern whites want to focus their production on cotton. Furthermore, Northern manufacturing supplies Southern most of the demand for finished goods up to and including cotton gins. Northern shipping carries the cotton to the mills of Europe, with the profits for transport going into Northern coffers. These are not trivial investments into slavery by any means and should cause us to abandon any notion that a bright line exists between free states who have nothing to do with the institution and slave states famous for having everything to do with it. All through the antebellum, a powerful contingent of Northern leaders preach concessions and accommodation for slavery. Some of them even own people themselves, including two senators, and derive direct profits from enslaving them.

That's a whole lot of responsibility, but is it as much responsibility as the white South had for slavery? With very few exceptions, Yankees don't own people themselves. They don't care where their cotton comes from or what their own products help sustain and that carries with it a fair degree of culpability, but they had the option to continue slavery at home and they declined it, albeit sometimes quite narrowly. By the 1840s, though often viciously racist and not above trying to find ways to purge their states of black people of any condition, northern Whites in general and in every northern state had abolished slavery. None I've ever heard of sought to re-establish it. Rather they had decided, and would increasingly decide over the next 25 years, to view slavery as foreign, unwelcome, unhealthy, and unprofitable. It was a thing which they wished quit of, even though they remained committed to a Union that included it elsewhere.

The story is almost the opposite for the South. Slavery in a few states is in decline. Demographic and economic changes have drawn them in a somewhat more northerly direction, though never without controversy. Elsewhere, it remains healthy and roaring along to ever-expanding horizons. The story of the twenty years before the Civil War is one of almost constant Southern triumph. They secure, through shady means, a vast new enslaved territory in Texas. They get the promise of potential slavery in the Southwest and set to making that a reality. They scheme to either turn California into a slave state or divide it and get a slave state out of the bargain. They achieve unprecedented authority to compel Yankees to enforce slavery on Northern soil in the Fugitive Slave Act. They roll back a thirty year old ban on slavery by repealing the Missouri Compromise. They fight a low-intensity, small-scale war to bring slavery into Kansas against the will of the white people who moved there.

Southern success in prosecuting slavery against the white North gives credence to the arguments of antislavery Yankees and so drive previously indifferent whites into their ranks. They respond by forming a new political party, the Republicans, specifically to restrict slavery back to what they consider its proper bounds in order to preserve the republic (hence the name) and the northern way of life against the aggressive slavocracy. As a result, the GOP almost wins the election of 1856 and then does win in 1860. None of this is to say that the North lacks responsibility for slavery in the South, but it has a whole lot less responsibility than the people actually practicing it and working to expand it do even before they launch a rebellion in its defense.