This answer is making use of Daniel R Hicks' links in the comments of the question.
The Claim
Donald Trump's tweet's phrasing "Since taking office I have been very strict on Commercial Aviation." is, perhaps typically, not very precise. Asked about details, the White House released a statement quoted by factcheck.org:
“Last year, the President announced his initiative to modernize Air Traffic Control and under his leadership, the Department of Homeland Security released enhanced security measures to ensure safer commercial air travel,” said Raj Shah, principal deputy White House press secretary.
Actions from Trump's administration
The President's "initiative" is a proposal to privatize the Federal Aviation Administration’s Air Traffic Control system. It has not been implemented.
The security measures from Department of Homeland Security (DHS) are:
The Chigaco Tribune also notes that the Trump administration didn't push forward a pending "important aviation safety regulation" proposed by the Obama White House about shipping lithium batteries.
Possible impact on flights safety
Privatization of air traffic control
The privatization of the Air Traffic Control system has not happened to this date, hence it is fair to say it had zero impact on flights' safety in 2017.
Whether this eventual privatization would enhance security is at least a very long shot. According to the Chigaco Tribune:
Supporters of privatization have been clear that the proposal is aimed at speeding air traffic modernization and is unrelated to safety.
Measures from DHS
The measures from DHS aim "to prevent terrorist attacks". They concern only the flights into the US from nine airlines. They do are safety measures, but one cannot prove whether they have or haven't deterred any intended terrorist attack on one of those flights. There sure has been no such attack in 2017, but neither has there been any during the past decades (even the attacks on 9/11/2001 hijacked domestic flights that would not have been in the scope of those measures).
Speculations aside, factcheck.org politely notes that
The White House didn’t provide any evidence that the DHS security changes had any effect on the level of airline accidents.
Reuters reports :
Current and former aviation safety officials said Trump was not responsible for last year’s unblemished commercial passenger jet safety record, citing years-long improvements in safety and the fact that no U.S. passenger airliner has had a fatal crash since 2009. Aviation Safety Network President Harro Ranter, whose group tracks aviation incidents, said in an email: ”It’s impossible to link the worldwide level of safety directly to recent U.S. policy changes.”
and concludes with Bryan Cranston's tweet adressed to Donald Trump:
“Your active participation monitoring the flight patterns and safety regulations was greatly appreciated. In 2018 could you please turn your efforts toward preventing wildfires and hurricanes?”
(maybe this tweet is out-of-scope, but I like the humour)
Summing up
Trump's administration policy about flights' safety cannot be considered as revolutionary. Whether the few measures taken by DHS in 2017 can be qualified as "very strict" is subjective, but they most probably had no noticeable impact on the excellent safety record of 2017 (no death worldwide) that continues a years-long trend of improving aviation's security.
The White House administration has not provided any proof of such an impact that could justify Donald Trump's bragging tweet.
