It really depends on what job you are recruiting for. If you are making them complete a CRB check in the first place, then we can assume that it is necessary, or a legal requirement for the job. Why else would you go through the hassle of getting one done?
What is a CRB check for? To check whether potential employees have criminal convictions. Especially if they are working with children and vulnerable adults, if they have any record of abuse in that area. Also if they would be handling large sums of money for your organisation, CRB information will let you know if they have been honest – to date at least.
So if there is a legal requirement to have one, if it comes back with convictions on it, you have to decide how relevant they are to the job you would be asking them to do. If they have a conviction for drunk driving, are you asking them to drive as part of the job? Are they perfect apart from that? You could give them the chance, as the incident itself may be a few years back, different life circumstances, etc.
The short answer is that if it is a legal requirement that your staff have CRB checks, then if it comes back “not clear” then you are allowed to reject them for that reason. The “clear” CRB is usually a legal requirement when working with children and vulnerable adults.
If a CRB is not a legal requirement, then you need to be “reasonable” and decide whether the CRB information regarding their offence is going to be a problem in the job you are asksing them to do. E.g. a bookkeeper convicted for drunk driving is not so much as a problem as one convicted of petty theft/embezzlement