NEW YORK—Democratic presidential candidate Bernie Sanders on Tuesday ramped up the platform to overhaul Wall Street at the center of his campaign, proposing a set of measures that would go well beyond the 2010 Dodd-Frank law to rein in big financial firms and give more clout to consumers.

In a speech here, the Vermont senator reiterated his calls for breaking up banks that are too big to fail and to reinstate the Glass-Steagall Act, which kept commercial banks and investment banks separate. Reinstating the 1930s law would...