You shouldn't be concerned about his problems any more than he should be concerned with yours if you'd be going through something similar. If the business is going as normal then there's not much to think about, but if you notice a substantial change in his behavior or his ability to run the company, then sure.
Being in a startup company in general, however, can be tough, roughly one in ten actually makes it, the others collapse before becoming a stable company. You probably know this already though. If you notice an increasing threat to the company's health correlated to the situation then you must evaluate whether you can afford immediate suspension or not if everything goes south.
Knowing that, you should always afford to be unemployed for a few months, but you should have that backup regardless of the format of the company you're working for or its threats. Everyone can lose their jobs out of nowhere.
Prepare for the worst, hope for the best, but don't spend too much time or energy worrying, it never helps. Dale Carnegie has written a lot about the matter, you can influence situations for the better by preparing but never by worrying.
I'd be concerned about how your coworkers are able to access and "accidentally" read private emails though, but that's a different subject.