Bistamp is one of the most reliable Bitcoin exchanges nowadays. They are very well-connected in the industry. The company has attracted significant investment from Pantera Capital, one of the biggest Bitcoin-specific venture capital investors, owned partly by Benchmark, Ribbit Capital, and Fortress Investment Group. They also work with Coinbase and other popular exchanges.
In terms of security – it offers two-factor authentication and a password, which needs to be changed every 90 days.
The exchange was last time hacked in January of 2015, and the crooks have stolen up to 19,000BTC. More info can be found in this.
Bitstamp stores 98 percent of all coins in cold wallets, keeping the customer’s funds offline in secure vaults. The remaining 2 percent are held for a live trading on the exchange.
The interesting fact is that the Bitstamp was the first major trading platform to incorporate the best security practices in the industry such as a fully insured cold-storage, two-factor authentication, and mandatory confirmation emails in order to increase the security of the users.
Also, I personally use it for trading and I never had even the slightest problem with them, so Bitstamp is definitely a safe way to buy/sell and store Bitcoins.