Sabletodo's right - with a small point of clarification, or maybe it is a question.
On the fiddle is dishonesty associated with your job or your normal course of business. So, you might be fiddling tax, expenses, allowances, public money at your disposal, social security, ... Typically your target is your employer or the government.
I'm not at all sure that taking bribes would be on the fiddle? I'm even less sure about ordinary decent criminals. OK they may be on the fiddle as well as being bank-robbers, but the two are quite separate in my mind.
Edit - I've just seen "on the take". See my uncertainty above about taking bribes.
I think the quote from "Yes Minister" could be:
The Economy Drive: Ron Watson: What about drivers and transport service staff.
Jim: First to go, good Lord we waste a fortune on cars and drivers, and they're all on the fiddle.
As Panjandrum says, dishonesty associated with your job, and targetting your employer. Drivers claiming to work longer than they do, may be stealing petrol from the government, &c.