There is a framework for this kind of regulation in place already, in the form of money transmitter licensing.
Coinbase, for instance, is licensed as a money transmitter in 40+ states, and relies on those licenses to do business in most of the United States, including with regards to the issuance of USDC, per my understanding. Paypal also operates in the United States under this licensing scheme.
The rigor of money transmitter regulation and licensing requirements varies state by state, but regulators' rigor overall has been increasing steadily for decades. In addition to annual and quarterly paper filing requirements, a number of states—including California, New York, and Texas, among others—perform periodic (often annual) on-site examinations of licensed companies—paid-for by licensees—wherein visiting examiners expect to have complete access to records and data.
These examinations are rigorous. After an examination, it is very unlikely that a stablecoin issuer will be allowed to continue operating as a licensed entity without being financially sound, and without fully backing its financial obligations with safe investments.
Any inability by a licensee to comply with requests made during the course of an examination, and any inability to provide acceptable answers to questions asked, can result in fines, limitations placed on the operations of the licensee, or even the loss of the license. Depending on the jurisdiction, and depending on the specific activity, operating without a license may be a crime, and would likely put a licensee out of business.
Among the requirements imposed under these licensing schemes are bonding requirements, and "permissible investment" requirements. Typically, permitted investments include bank deposits and government debt, supplemented by surety bonds issued by large insurers. In many ways, these liquidity requirements are more rigorous than those imposed on banks. Most outstanding consumer liabilities—presumably including stablecoins issued by a licensee—must be backed by these permissible investments.
Of course, given that there are 50 separate state regulatory regimes with regards to money transmission, and given stablecoins' novelty, regulation specifically pertaining to stablecoins is not standardized yet. This is a matter of education and standards setting among regulators, however. It is not a matter of requiring new legislation at the national level, or even at the state level; nor does it necessarily require major involvement by the federal bank regulators. In most US states, banking regulators already have sufficient legal authority to craft stablecoin regulations if they want to—and they have the operational capacity to enforce such regulations as well—based on their authority to license and regulate money transmitters.