The company was fined £4,000 with costs of £3,289, after it pleaded guilty to breaching health and safety and use of work equipment regulations.
A 40-year-old employee, who asked not be named, was clearing jammed booklets from a stacker on 4 May 2009 when the incident occured.
During a Health and Safety Executive (HSE) prosecution at Banbury Magistrates Court, it was described how the stacker used belts that ran over rollers to move the booklets along to be packaged for delivery.
While the worker's hand was under one of the belts as he attempted to remove booklets, the machine started up and caught his left thumb between a feeder belt and a roller. His thumb was pulled under the roller and then crushed and cut.
The man required both internal stitches and external stitches to his badly lacerated thumb and was off work for six weeks.
According to the HSE investigation, a guard was missing from the machine, which would have protected the rollers and prevented access to the moving parts.
Since the incident, the company has installed a guard over the roller.
HSE Inspector Sharron Cripps said: "This incident highlights the dangers associated with cleaning and maintaining printing machines. It is important when undertaking a risk assessment to consider less frequent activities such as un-jamming blockages, as in this case.
"This incident was avoidable, and had this type of work been properly planned, then this would never have happened."
BGP was unavailable for comment.