Of Australian-English origin, the phrase to keep the bastards honest means: to ensure that politicians, large business organisations, etc., behave fairly and openly, and are accountable for their actions.

This phrase occurs, for example, in the following letter to the Editor, published in The Independent (London, England) of Friday 25^th September 2015—“the behaviour of […] VW” refers to the Volkswagen emissions scandal of 2015:

Corporations are by nature psychopathic. Their purpose is to achieve the maximum return on capital for their shareholders, and nothing else. The interests of customers, employees, regulators and the environment are obstacles that have to be negotiated and minimised in pursuit of that aim. This may be done legally or illegally.
Once this is accepted, the behaviour of the banks, oil companies and now VW makes perfect sense. Deceiving the customer has been going on at least since merchants adulterated flour with chalk and watered their beer. You have to keep the bastards honest.
Nigel Scott
London N22

The phrase to keep the bastards honest was coined in 1980 by the Australian politician Donald Leslie Chipp (1925-2006), Leader of the Australian Democrats. The following explanations are from Australian Democrats: the passing of an era (Canberra: Parliamentary Library, 2009), by Cathy Madden:

The party was formed in May 1977. The Australian Democrat party had its genesis in two existing parties: the Australia Party (formerly known as the Liberal Reform Group) centred in Victoria and the New Liberal Movement based in South Australia. Discussions had occurred between the two parties about seeking an alliance if not amalgamation. The two groups found a common basis for a new political movement, in the widespread discontent with the two major parties following the constitutional crisis of 1975. In the former Liberal Government Minister, Don Chipp, the two groups found a leader and spokesman.
[…]
Following a series of well attended and successful public meetings to build party support, Don Chipp was able to bring together disillusioned segments from the major parties and other groupings: the party was born.
Don Chipp’s 1977 election campaign was centred on a return to the political virtues of ‘honesty, tolerance and compassion’. He described the new party as an independent, incorruptible, middle-of-the-road ‘watchdog’ for the Senate.  It was during the 1980 election campaign that Chipp voiced the famous catchcry of the Democrats—that they would ‘keep the bastards honest’ *. The ‘bastards’ in this context were the major parties and, more significantly, the government of the day.
[* D. Chipp, press conference, 19 September 1980.]

However, the earliest occurrence of the phrase to keep the bastards honest that I have found predates Don Chipp’s press conference of Friday 19^th September 1980; it is from Singers come to the aid of the party, by Aileen Berry, published in The Age (Melbourne, Victoria, Australia) of Tuesday 29^th April 1980:

“We’ll keep the bastards honest, we’ll never block Supply.” With that emotive cry, the Australian Democrats’ most unusual piece of political merchandising was launched in Melbourne last night.
The party’s offers of cut-price grog, cheap travel and insurance, and do-it yourself conveyancing kits faded from mind, as the new singing group, the Chippunks, made its first public appearance at a Democrats’ rally.
The founder of the Australian Democrats, Senator Don Chipp (after whom the group was named) listened as the tale of the “sturdy Democrats, stout and holy knights in white” unfolded.
To the tune of ‘Waltzing Matilda’, the Chippunks continued: “Siddons for Senate. Siddons for Senate, Who’ll come a’voting John Siddons with me.”
Mr Siddons, the man who holds the number one spot on the Democrat Senate ticket, sat behind the singers, alternately beaming and blushing as they sang his praises.
To thunderous applause, the Chippunks closed with: “And they sang as they cheered, ‘Now the nation’s right again’, Siddons in Senate for you and for me.”
More than 200 supporters of the Australian Democrats gathered at the State College, Burwood, to hear Senator Chipp and other speakers discuss industrial conflict in Australia.

The second-earliest occurrence of the phrase to keep the bastards honest that I have found is from Democrats to be honest brokers, Chipp promises, by Paul Ellercamp, published in The Sydney Morning Herald (Sydney, New South Wales, Australia) of Tuesday 23^rd September 1980:

The Australian Democrats will “keep the bastards honest,” if they win the balance of power at the Senate election next month, the party’s leader, Senator Don Chipp, promised last night.
He said the Democrats would vote against Government legislation which they adjudged broke campaign promises.
And they would never vote to block Supply in the Senate to bring down a Government.
[…]
He said all Democrat candidates had signed statutory declarations promising never to vote against Supply, following support for the proposal from party members in a national plebiscite.
And they would allow no legislation through the Parliament breaking campaign promises.
Senator Chipp placed great stress on the second commitment, because he said it would prevent incidents such as the Fraser Government’s removal of half-yearly pension indexation, soon after an election promise to maintain it.