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The phrase the strong arm of the law, and its variants, designate the far-reaching, inescapable or punitive power and influence of the law.—Synonym: the long arm of the law.

—Note: More generally, the noun strong arm has long been used to designate far-reaching power and influence. The following, for example, is from Religio Stoici (Edinburgh: Printed for Robert Brown, 1663), by the Scottish author and lawyer George MacKenzie (?1638–1691) [The Virtuoso, Or Stoick, page 36]:

Christianity […] gives a check to presumption, and suffers not man to think himself the sole arbiter of his own condition; because God can easily quash these babylon-like fancies, which his topless ambition is still a building; and to his despair, because a lift from the strong arme of Providence, may heave him up above all his difficulties.

These are, in chronological order, the earliest occurrences of the phrase the strong arm of the law and variants that I have found:

1-: From the Sheffield Register, Yorkshire, Derbyshire, & Nottinghamshire Universal Advertiser (Sheffield, Yorkshire, England) of Saturday 16^th February 1788 [page 1, column 4]:

HOUSE OF COMMONS.
Feb. 8.
Royalty Theatre.

Mr. M. A. Taylor moved for leave to bring up a petition from John Palmer, and others, the proprietors of the Royalty Theatre, setting forth that they had, on permission from the Constable of the Tower, and in the belief that that permission was a legal and sufficient authority to enable them to act plays within the limits of the Tower hamlet, expended a considerable sum of money in building a theatre; and praying for leave to bring in a bill, on which his Majesty might grant them a patent for that purpose. If this petition should be received, another in support of it would be presented, signed by the principal persons in that part of the town.
Mr. Anstruther said, that for twelve months past the proprietors of the Royalty Theatre had been setting the laws of the country at defiance, and now that the strong arm of the law had reached them, and they were no longer able to elude the coercion of the King’s-bench, they came with a petition, praying Parliament to legalize all the illegal acts that they have been committing. He conceived there would be little doubt in what manner such a petition ought to be treated by the House.
The Speaker put the question, and the petition was rejected.

2-: From The Chester Chronicle; and General Advertiser (Chester, Cheshire, England) of Friday 19^th June 1789 [page 3, column 4]:

Mendoza, the bruiser, has modestly annexed P. S. P. to his name.—Many of the literati have conceived, that he has been elected an honorary member of some of our academies; but the most probable conjecture is, that it means no more than—Professor of the Science of Pugilism.
The good fruits of the fist are likely to extend over the whole English soil of civilization.—Three Warwickshire lads have challenged Johnson, Mendoza, and Humphreys, by public advertisement, to single combat, for any sum they please.—It is to be hoped, that this polite practice will at last receive a blow from the strong arm of that formidable bruiser—Law!

3-: From Observations on the Frauds, committed by the Bakers of the City of Dublin; With some Hints at the proper Modes of Prevention as well as Punishment; In a Letter addressed to Travers Hartley, Esq. M. P. (Dublin: Printed by Zachariah Jackson, for Messrs. Grueber and M‘Allister, 1789) [page 14]:
—Notes: This letter, dated Tuesday 11^th August 1789, was signed by a “Friend to Humanity”. Travers Hartley (1723-1796) was an Irish merchant and Member of Parliament for the City of Dublin:

Thus is the iniquitous Trade carried on through all its Branches with Impunity, in open Defiance of every Law Human and Divine: and though the Bakers may now almost plead a prescriptive Right to it, yet the length of Time they have been in the peaceable Possession of their Iniquity calls but the louder for the strong Arm of the Law to cut it up by the Roots. Vain is every puny Effort; and nothing but a Law, or a Regulation that will strike at the Foundation, can ever overturn a Superstructure that Time and barefaced Villany [sic] have but too firmly erected.

4-: From an account of the debates that took place in the Irish House of Commons on Thursday 19^th January 1792—as published in The Northern Star (Belfast, County Antrim, Ireland) of Wednesday 25^th January 1792 [page 1, column 1]:

Mr. Knox […] made some remarks on the anarchy and confusion which had for some time reigned in France, and had been, he said, fomented by the influence of speculative learning and by desperate men, to whom the blessings of peace was Pestilence, and the beauty of order Deformity. Such spirits there were in this country, but too contemptible to attract apprehension, or to be dragged from their lurking holes: but to such he would say, that if their turbulences fermented into sedition, they would raise against themselves the strong arm of the Law and be crushed.

5-: From The Bath Chronicle (Bath, Somerset, England) of Thursday 29^th November 1792 [page 4, column 1]:

Thursday a motion was made in the Court of Admiralty, on behalf of the India Company, to be heard by counsel against a decree of the High Court of Appeals, in the long-litigated cause between the Company and the crew of the Nymph sloop of war, relative to the capture of the Dutch fort of Chinsurah, where property to the amount of half a million sterling fell to the captors, and which had been directed by the Court to be divided between the sloop and the Company’s troops. Sir James Marriott reprehended the mover, and insisted on an obedience to the order of the Court; he said, when an attempt of that nature was made to sport with justice, it was high time the Courts should stretch forth the strong arm of law in justification of individuals.