Which Beatle came from the poorest background?

All four Beatles had what Mark Lewisohn calls ‘unvarnished working-class roots in an industrial city that had seen better days.' 

The Everton area of Liverpool, 1960

That said, they grew up in an era of full employment and rising living standards ('You've never had it so good'). Low-paid work was widely available.

None of the future Beatles experienced parental unemployment.

Austerity

The new National Health Service meant that medical care was free at the point of delivery, with the famous exception of 'teeth and specs.' Working-class people also had widespread access to high-quality education. John, Paul and George all went to high-performing state schools in adjacent areas. 

The immediate post-war years were technically more affluent, but later came to be seen as a time of austerity. Rationing, which continued into the early 50s, was introduced to ensure everyone had access to enough meat, vegetables, clothing, etc.

A world in black and white, as it would later be described

John

I moved in with my auntie, who lived in the suburbs in a nice semi-detached place with a small garden, and doctors and lawyers lived around us... not the poor, slummy kind of image projected in all the Beatles stories. John Lennon

Only during his earliest years did John experience anything approaching poverty.  In his final major interview with Playboy, he describes in detail how his upbringing differed from his bandmates:

In the class system, it was about half a class higher than Paul, George and Ringo, who lived in government-subsidised housing. We owned our house and had a garden. They didn't have anything like that.

Paul's first impression of John's family was that they were 'very posh'. McCartney didn't disapprove. The new social rules seemed like a logical extension of the ones his mother had taught him.

One immediate class signifier Paul noticed was John's use of 'Aunt Mimi' rather than the more familiar 'auntie'. Interestingly, John adopts the word 'auntie' in the interview.

Paul and George

As Lennon points out, Paul and George (but not Ringo, see below) grew up in council houses (social housing), though neither family considered themselves poor. 

Paul's family was notably aspirational. His father, Jim, was a cotton salesman, and his mother was a nurse and midwife. George's parents were a fraction lower in the then-social-class hierarchy: his father was a bus conductor (and later driver), while his mother worked in a shop.

Aunt Mimi noticed a subtle difference in accent and manners ('George is very 'dose', John' she noted disapprovingly, a reference to his 'scouse' working-class accent). This snobbery stemmed from social insecurity; Mimi did not like being reminded of her own humble start.

Ringo 

Tellingly, John mistakenly implies Ringo came from a similar social background to George and Paul. Richard Starkey had, in fact, experienced absolute deprivation, growing up in Dingle, an area notorious across the city for its poverty and crime. 

The Starkey family had no access to the relative luxuries council houses offered. They paid ten shillings (£0.50p) a week to a private landlord for 10 Admiral Grove, a terraced house without a bathroom or indoor toilet.

The house where Ringo lived. Copyright Pernille Eriksen — reprinted with permission — prints available

Ringo was the outlier of the future Beatles, experiencing 'a Dickensian chronicle of misfortune' in a childhood combining poverty, ill health, poor education and paternal abandonment. 

Ringo's Childhood: 5-minute free read on Medium