Yes, Klopp is still safe at Liverpool because he is a coach who has the respect of the fans of his team and club. Klopp needs a lot of support to improve Liverpool's current form and morale, but that will soon be different when they start the competition again next week. Chelsea were impatient with Tuchel, they did not learn from Arsenal who kept Arteta despite Arsneal's poor performance.
Now Klopp's task is certainly difficult considering Liverpool do not have good players to navigate this tight competition. They lost a few players through injury, and that has cost them some strength against big teams like Manchester United and Napoli.
Liverpool is at a crossroads right now, with the level of poor results that the club have hard this season both in the premiere league and champion league Liverpool needs a lot of cooperation between both players and coaches as that is the only way to improve the present condition of the club, having to fill the gap mane left behind that is effecting the club Liverpool spent a huge amount of. money is buying players in the last transfer window so I don't see any reason why the club is still underperforming at this moment.
And on the issue of Chelsea the sack of the coach was an unexpected action from the club management and this time was bad timing for such an action and this will cause a drawback to the club's success this season.
So both Chelsea and Liverpool will be facing a tough time this season and in the next coming weeks a lot will change but in which direction that change will be we can't tell for now only time and clubs' efforts will tell.