Hi, try to do the followings to determine:
1. team and company visibility - make sure that the team and company profile are sufficiently highlighted, this includes business address and linkedin profile.
2. safety - trusted exchanges have 2-factor authentication as an option. I would frown upon any sites that don't offer this feature.
3. trading volume - In theory the bigger the better. The bigger the trading volume the more reputable but an easier target for hackers. Coinbase, bittrex, kraken, bitfinex are site with heavy trade volumes.
And to be honest, anyone that trades heavily on cyrptocurrency should always store their coins in their own offline computer or hardware wallet. This means having the private key only to yourself, this is the only way to proof you *REALLY* own the coins.
Remember, at the end of the day, many reputable sites get hacked and there is no strong government oversight yet. If you are just starting out I'd say don't sweat it and keep your coins in the exchanges until you are comfortable with how things work in the cryptoworld, slowly as you increase your holdings or go long term, then move your coins somewhere where you can own your private key.
PS: If you can share which site you are referring to maybe someone can offer some experience.