The virtual currency bitcoin is getting a very real boost on Monday, with the opening of the first licensed U.S. exchange. Coinbase Inc., a startup backed by $106 million from the New York Stock Exchange, banks and venture-capital firms, said its exchange will offer greater security for individuals and institutions to trade bitcoin and monitor real-time pricing of the cryptocurrency. The exchange could bring needed legitimacy to the currency, which isn’t backed by a central government and is traded over virtual exchanges, primarily overseas. Coinbase said it has insurance, offering traders some assurance that their money won’t disappear. Bitcoin enthusiasts have been buffeted by the collapse of Japan-based exchange Mt. Gox last year — taking with it around half a billion dollars of investors’ money — and a security breach earlier this month at Slovenia-based exchange Bitstamp. The value of a bitcoin itself, determined by trading on existing exchanges, has fallen to about $240, from a peak in late 2013 of more than $1,200. Coinbase’s founders say they have been working for five months to win licenses from state financial regulators. They have regulatory approval in half of U.S. states, including large population centers like New York and California. For now, Coinbase can do business with account holders only in states where it has approval. An expanded version of this report appears on WSJ.com.