My PI is irresponsible and I worry about my career as a postdoc I am a postdoc in life sciences in the US and have started my postdoc around 6 months ago.
I am at a great (top-10) university. I like the university, lots of smart people and friendly environment.
I love my work and plan to stay in academia. I am from europe and have left home just for the postdoc.

I am, however, very worried because my PI does not seem to bother too much about our career prospects.
Some points:

He is often away from the lab due to family-related problems. He spends a lot of time with his family (has 2 kids). Although that is great, he does not spend much time in the lab, forgets meetings, is not up to date with current research and often does not have time for supervision (this was the case even before the pandemic). He always comes up with some excuse but after a while it has become clear that his priority is just very much somewhere else than research. 

Currently, the lab is supposed to write a grant. He has left that to us, he says that he is too depressed because of the current situation with COVID-19 to be able to focus on work.
This is very worrisome, because our future is dependent on the grant, yet at the same time
he wants us postdocs to go back to the lab as soon as possible (presumably to keep face and show the university our lab is active, while at the same time he does not seem to care about what happens to us postdocs).

In addition, we have currently submitted a paper (~2 years of work from the lab) which gets rejected from all top-tier journals.

He has a lot of good properties, he listens to our opinions, accepts criticism, leaves us a lot of freedom. As mentioned the university is great too. I am highly motivated and love research, but
worry extremly that I am just wasting my time. I have my own money for now but can't change labs easily and would prefer to stay at the uni.

What should I do? 

Edit I do understand that as a postdoc it is important to be independent. I am. I have learned several relevant skills, have my own ideas, I am generally quite pro-active. But typically as a postdoc you can also join an environment which allows us to find your own goals within the work of the lab. My PI is just so out of touch with everything, that his ideas are completely outdated. He just does not take research seriously (even though he still wants me to publish asap again under the current circumstances). Hence he can't also give me any guidance (he suggests experiments which have been done many years ago already) and, worse, does not even bother to change that.

Edit2 The PI just fired the other postdoc in my lab. I am under a complete shock. Basically a mix of blaming the other postdoc for not advancing with the grant proposal (PI promised many times to finally get it together and work on the proposal) which was never really officially discussed as being our task to begin with. Yet he has the audacity to whine during the "crisis" meeting with me about how sad his family life is (his kids can't go to school which they don't like yada yada) and then to interrupt the meeting to go and play basketball with his kids. I have told him now that I want to have a co-supervisor and he has promised to reach out to people. He is so reckless, selfish, if he does not get it together I will change labs even if I lose my fellowship.
Of all you say, the only indication I see of irresponsibility is his desire, you say, to send you back to the lab before it is safe. The rest is understandable. 

We live in perilous times. He may be entirely correct in focusing on his parental duties and may have no other option with child care essentially non existent in many places. Many people are depressed now, but it isn't a sign of lack of responsibility. "Forcing" you to write a grant can be a boost to your career and he might have wanted to do that in the best of times. 

Not getting accepted into the top journals can be pretty common and may not be his fault, or at least not entirely. 

And yet you say he has many good qualities. 

As to what you should do, I recommend that you maintain social distancing for your own safety. But beyond that, just carry on as best you can and hope that the horrors of the current situation don't increase or become cyclic. Make sure that you don't burn bridges with your professor, as that will make your next career steps (after the pandemic) much harder. 

Encourage the professor to deal effectively with depression if you can, and deal with your own if you feel those twinges. 

You are in a bad situation. A worldwide pandemic tends to cause that. It is no consolation, but in other times and places it was much worse. 



Another consideration you might make as to the level of his responsibility is how helpful he is at getting postdocs their next position. If he gives good support and they wind up in good positions, then you can put up with some things that wouldn't make sense otherwise.