A safe pair of hands
A reliable, if somewhat dull, person who can be entrusted not to make a mistake with a task.
UK origin. Applied to politicians or diplomats who were given sensitive work that required careful handling.
The earliest references to the phrase link it to the public school sports of cricket and rugby. It's easy to see how it migrated from there to the public school dominated world of Victorian UK politics. These two authors both included the term in tutorials on sport:
James Pycroft The Cricket Field, 1851: "The safest pair of hands in England".
W. J. A. Davies How to play Rugby Football, 1933: "A safe pair of hands is of paramount importance."