Following decades of corrupt and brutal dictatorships, South Korea takes a relatively tough view of the abuse of power and corruption by the president.
There is also a strong case to be made that the living ex-presidents were worse than typical presidents in healthy non-corrupt democracies.
The earlier two of the ex-presidents, Chun Doo-hwan (the last of the military dictators) and Roh Tae-woo (the transitional president between dictatorship and democracy; he had been selected by the dictatorship to be the next president, and popular protest forced the government to accept a new democratic constitution, under which Roh was elected as the next president) were sentenced to death and 17 years respectively for their roles in the Gwangju Massacre. Quite reasonably, democratic Korea chose to treat killing thousands of pro-democracy protesters as a serious offence. Chun was president at the time of the massacre, and bore the chief responsibility (thus the death sentence). Roh's role was as Commander of the Defense Security Command. The charges against Chun and Roh included mutiny (for their role in the coup that overthrew the hoped-for transition to democracy after the assassination of long-term dictator Park) and treason (the Gwangju Massacre) and corruption. They only served a few years, before being pardoned in the interests of national reconciliation.
The latter two of the ex-presidents sank in a swamp of corruption and abuse of power. Park Geun-hye's fall is recent and well-publicised. The Korean people's attitude to the revelations of corruption and hew abuse of power are well-shown by the drop in her approval rating from a high of 63% to below 5%. Her conviction and long sentence are a sign that South Korea is interesting in protecting its hard-won democracy. Her predecessor, Lee Myung-bak, greatest offence was taking money in return for a presidential pardon. The South Korean people - and the court - viewed this action, pardoning a criminal in return for financial support, as deeply immoral and acted accordingly (and since it was also illegal, could do something about it). Hwang Kyo-ahn, Park's prime minister (selected to replace her original prime minister after he resigned amid allegations of bribery, making it rather reminiscent of Ford replacing Agnew and then Nixon) who took over as acting president after Park's impeachment, and can be counted as an unofficial ex-president, avoided the fate of his two predecessors, one hopes by avoiding corruption and abuse of power (at the very least, he was not caught).
There were three other democratic ex-presidents, between Chun and Roh, and Lee and Park. All are deceased, so aren't living ex-presidents. The first of them, Kim Young-sam (president 1993-1998) brought in strong anti-corruption attitudes and measures. The next, Kim Dae-jung, had been sentenced to death (commuted to 20 years due to foreign pressure) under the Chun presidency. His crime: supporting democracy and opposing a dictatorship that killed thousands of its citizens. The third, Roh Moo-hyun, committed suicide while under investigation on charges of bribery. In his earlier career as a lawyer, he had been briefly imprisoned (AFAIK, he was released without trial) - supporting human rights against a dictatorship can be dangerous.