I like the massive pandemic setting. I like that the US government becomes the Speaker of the House in a bunker, and then probably nothing. I liked the Cruise Ship of Death, though I was wondering for a while where all the people were. That a crew member would lose his helmet and get exposed was completely predictable, but not so bad.
I liked them witnessing a nuclear attack, though we don't know the motivation behind that one yet.
It's good to see Adam Baldwin again. He'll always be Jayne to me, but he's fun to watch.
However, it is a Michael Bay production, so The Last Ship had lots of stupid in it.
For example, if you're a rogue Russian military squad out to capture The Cure, maybe, just maybe, shooting at the people who know how to make the cure is a wee bit counterproductive. By which I mean the scientists collecting the vital data, not the crew of the destroyer in general. And since we've established that it was a 24-grade mole who tipped them off, they should have known full well that they're still working on The Cure, weren't remotely close, and that interrupting them was not a good idea.
It was a bit much that 3-4 Navy types on snowmobiles could defeat a much larger force with attack helicopters. The Russian troops clearly went to Stormtrooper marksmanship school.
"Radio Silence" does not mean "we cannot receive anything." It means you don't transmit. Keeping the DD's crew in the dark that way served no purpose, but more significantly, there's no way they could have sold "don't listen to any broadcasts" to the captain in any plausible way.
Holding a fuse in by brute force does not prevent the fuse from blowing. It doesn't help a damned thing, actually. If the fuse is blowing, there's a reason it's blowing. Fuses are sacrificial parts meant to prevent more serious damage. If you defeat the fuse, you're probably causing an electrical fire instead. The circuit doesn't "get better" once things start again.
"We needed the primordial virus" is of course bullshit. Not the dumbest science-related thing you see in most TV SF, but dumb.
The unnecessary conflict between the captain and the XO was predictable, but also annoying because the XO's position is too obviously idiocy. That part reminded me uncomfortably of Last Resort, that show about the SSBN going rogue that was much dumber than it needed to be. Though in that case, it was generally the captain with the dumb position, not the XO.