This should be titled 'When to join an externally funded startup'.
It's a whole different ballgame if you join one that's actually making a profit and doesn't need someone else's cash. Get in as early as you can, if you like the idea and can see it's got longevity.
I think you're referring to a business, not a startup. The definition of "startup" has been tranformed into basically "a business that is primarily based on a website or mobile app," which it is not.
Startups rely on fast growth. If you have a very narrow market or your end goal is $200k/yr in profit, you are by definition not a startup.
If you're actually a startup, you should not have any profit in the beginning. You should be pumping that money back into the company and seeking external funding in order to pour gasoline on the fire.
There's nothing wrong with having a business instead of a startup (personally I prefer the bootstrapped business route) but to say that it's a startup is not accurate.
This is true, and is often overlooked. Many startups are internally funded, and grow through profits. These companies are typically more conservative in their business strategies, because people tend to have less risk tolerance with their own money, because there are no outside investors pushing for accelerated growth, and sometimes because of more limited finances.
It's a whole different ballgame if you join one that's actually making a profit and doesn't need someone else's cash. Get in as early as you can, if you like the idea and can see it's got longevity.