As usual with these issues the answer is never quite as simple and an employer would like.  The simple bit is that if your errant member of staff has a job that requires them to drive and if the driving conviction means that they are banned from driving, then you have grounds to dismiss them if you wish because they are unable to do their job.  This would apply to the travelling sales person, or the delivery driver or similar job.  If you don’t want to dismiss them then you will have to make sure they can do the job some other way, or change their job specification to get them doing something else.

If their job doesn’t require them to drive and they can still get themselves to work then there may be no need to dismiss them.  

But if your employment contract states that committing a criminal offense is grounds for dismissal then you would still have a reason to dismiss them.

Even then it is also possible to dismiss someone for their conduct outside of work if it has a bearing on their job i.e. if it “affects the employment relationship”.  For example, in the case of drink driving if the person was training or managing people who were drivers for the company you could make a case that it was setting a bad example within the company and so was grounds for dismissal.