I'll assume that you're in England or Wales. (Things are slightly different in Scotland and Northern Ireland. See below if that's where you are).
In England & Wales employers aren't generally allowed to check out the criminal records of their employees. They're only allowed to do so when the type of employment falls within one of the 66 categories which are exempted from the provisions of the Rehabilitation of Offenders Act. That generally means work with, or alongside, children or vulnerable adults, or with the police, courts and security services. The categories are listed here:
http://www.homeoffice...ons-guide?view=Binary
When applying for jobs within MOST areas of employment (where CRB checks aren't available to employers anyway) criminal convictions become 'spent' after a certain period of time, which is defined by the sentence imposed by the court. See here for details:
http://www.nacro.org....cro-2007021302-65.pdf
However anyone (in England & Wales) applying for a job where a CRB check is available to employers must be within one of the 66 categories referred to above, where convictions are NEVER regarded as spent. They will ALWAYS show up on any CRB check.
England and Wales only have 'standard' and 'enhanced' checks. As stated above, they are only available for certain types of employment and always show all convictions, however old they might be . Scotland and Northern Ireland have a lower ('basic') level of checks, which are available to all employers and only show unspent convictions.
So, depending where you are (and whether your reference to a 'basic' check meant a genuinely 'basic' one or a 'standard' one) your convictions will either always appear on any criminal record check or will disappear once they're 'spent' (unless you're then applying for work in an 'exempted area' of employment).
However there is no way that they can ever be deleted. (Well, not before your 100th birthday, anyway!).
Chris