When it comes to hiring, employers can't discriminate on the basis of age, race or gender, but there's nothing in the constitution that specifically protects applicants with a criminal past.
Even so, there is some constitutional basis for the "ban the box" laws. (The box is the box on applications that candidates must check to indicate if they have a criminal record. These laws typically allow background checks later in the hiring process, perhaps after a first interview or after a conditional offer.) The constitutional basis comes in the concept of "disparate impact."
This means that sometimes a policy that seems race or gender neutral has the effect of discriminating, even if that wasn't the original intent of the policy. If men tend to have more prison terms in their past and if an employer has a blanket policy that prohibits anyone with conviction from being hired for a job without consideration of any other factors, that employer may end up discriminating against male applicants. That would make the employer in violation of civil rights rules enforced by the U.S. Equal Employment Opportunity Commission.
So what does the EEOC suggest? More careful consideration. Among the factors: Does the conviction relate to the job? For example, a person with a conviction for bank theft would have no reasonable expectation of getting a job as a teller. When did the conviction happen? Does one bar fight as a rambunctious 22-year-old mean that the same person, now a 52-year-old church deacon, should not be hired to work in a nursing home? Has there been evidence of rehabilitation? If that 22-year-old spent the next 30 years with a clean record and a history of community service, shouldn't that behavior outweigh one youthful indiscretion?