WASHINGTON—The Securities and Exchange Commission will regulate cryptocurrency markets to the maximum extent possible using its existing authority, Chairman Gary Gensler said Tuesday, while also calling on Congress to grant the agency more scope and resources to oversee the sector.

Calling the asset class rife with “fraud, scams and abuse,” Mr. Gensler signaled the SEC is likely to become more active in policing crypto trading and lending platforms, as well as so-called stablecoins.

“We just don’t have enough investor protection in crypto. Frankly, at this time, it’s more like the Wild West,” Mr. Gensler said in prepared remarks to the Aspen Security Forum. “We have taken and will continue to take our authorities as far as they go.”

U.S. financial regulators have struggled to get their arms around the fast-growing world of cryptocurrency and related financial technologies. Unlike in the securities and derivatives markets, no single regulator oversees crypto exchanges or brokers. As the market value of the asset class has exploded to more than a trillion dollars, so have scams.

He said large parts of the sector operate outside of regulatory frameworks that seek to protect investors and consumers, reduce crime, promote financial stability and protect national security.