Looks safe and trusted based on this reputation review:
http://www.scamfoo.com/site/bitfinex.com/Most important facts:
1) The domain name was registered 4 years ago.
2) This site has a very good online popularity (38K Alexa rank)
3) This site uses an HTTPS secure connection (by GeoTrust Inc).
4) This site has an excellent (82/100) trustworthiness score.
5) The site is not blacklisted by any service
Do you know they had been hacked in early August 2016 and lost almost 120k bitcoins?
Regarding trust, you should not trust any exchange by default. So if you have to keep your funds there, keep only as much as you actually need for trading. Regarding Bitfinex specifically, I don't think they will ever regain their trust (what's lost is lost), this is a tough business and competitors are not sleeping. Look at what happened to Bter which had once been one of the largest altcoin exchange out there. Now it is a half-man half-exchange with fictitious volume and frozen order-books