Yes, it's perfectly safe to drink and extremely delicious. In The Philippines, we often call this 'Kopi Alamid', and it truly is one of the rarest coffees on the planet. At around $20 per 75 grams (local price), it's also one of the world's most expensive coffees.
The palm civet is a nocturnal cat (well, more like a ferret, but technically a cat) that eats the berries (beans) and is extremely picky in its diet. For the civet to choose a bean, it has to be just right when it comes to the aroma, and given the sensitivity of this creature's nose, said aroma means the bean is perfect. Some say the digestive process itself lends to better flavor, it can be a matter of heated debate depending on where you ask ;)
The thing is, the cat doesn't digest the whole bean, just the outer pulp. Farmers follow them around and wait for nature to take its course, then the beans are processed in a very safe manner. Farmers typically get just a kilo or so a day, it's quite an intensive process.
I drink it as often as I can afford to do so, and I have never gotten sick. It's a partly fruity, partly nutty and just an all over cozy and relaxing taste that I highly recommend if you're able to get the coffee. Just be prepared to pay quite a bit, and ready to consume it quickly after getting it home (it does not keep as well as others, probably due to local transportation issues prolonging the time it takes from roasting to actually get to market).
If you have a hand-powered burr grinder, I recommend using it and grinding pretty gingerly, even the heat of most ceramic grinders is enough to lose a little bit of the fruity taste you'd otherwise get. Really, treat these beans like royalty, they're that good.
If you buy some, try to avoid buying from farms with captive civets - the growing popularity of this coffee has led to a cottage trapping industry, which is (unfortunately) likely to seriously endanger them soon.