WASHINGTON—Coinbase, a leading cryptocurrency firm, has approached U.S. regulators about registering as a licensed brokerage firm and electronic-trading venue, a move that comes as regulators have waged an aggressive campaign to supervise the fledgling industry.

The San Francisco-based startup, one of the world’s largest platforms for trading bitcoin and other cryptocurrencies, has met with Securities and Exchange Commission officials in recent weeks about registering its business with the agency, according to people familiar...