No, you cannot DBS (formerly CRB) check a customer
DBS checks can only be carried out within an employment or volunteering context, provided specific conditions (such as the profession, work, or type of job) are met. It is not possible to lawfully conduct a DBS check on a customer or supplier, for example - because they are not an employee or a volunteer for your organisation.
It is possible for members of the public to request a DBS Basic check for themselves (this will reveal current active convictions), but not for other people. The other DBS checks are only available to organisations with specific hiring needs as mentioned above.
DBS checks are regulated by the Rehabilitation of Offenders Act 1974 and the associated Exceptions Order 1975. A DBS Basic check will reveal active, "unspent" convictions. A "spent" conviction is simply one that no longer shows up on a DBS Basic check. A DBS Enhanced check will reveal all convictions (active and "spent") plus relevant information from police forces depending on the nature of the proposed employment.
Most occupations that work with children or vulnerable adults in some capacity (e.g. lawyer, teacher, nurse, care home worker, school janitor, etc.) will be eligible for a DBS Enhanced check. All other jobs will only be eligible for a DBS Basic check.
There is no legal way for you to find out if a prospective customer or supplier, etc. has a criminal record in the UK.