It's mostly terrible. Mimicking phrases in a language you can't speak is incredibly difficult—especially if it's a language with a completely different set of phonemes and tonal qualities than what you're used to.
It's similar to when you hear non-English speakers singing English songs. Because you don't know what you're saying, even if you have the phonetic spelling written down as a reference, you're likely to screw it up when you try to deliver it naturally. You have the choice of, either:
- pronouncing it slowly, deliberately and accurately, or
- mumble through the line quickly to give a phonetic facsimile that has the appearance of fluency, but utterly butchering the actual words
For the most part, the actors seem to go with the second option, which is understandable given the context. Most of the phrases aren't meant to be understood by anyone. They just add color to the dialog.
But, yea, it's really, really bad, and most of the actors seem to know it. The intonation always feels slightly off, which is predictable since that's the hardest part for Westerners to adapt to.
It would have been nice if they'd received Mandarin lessons and shot part of the series in Taiwan, where they'd be interacting with Mandarin-speakers daily and hearing it regularly. Or, at the very least, it would've helped if they'd learned the Chinese phonetic alphabet. It'd have reduced the feeling that they were doing linguistic yellow-face.
But you really can't expect that much from an American TV show. At the very least, Whedon tried an interesting idea, and his Mandarin is reasonably accurate when he speaks it.