It was the congressional majority that repealed the Glass-Steagal act; Bill Clinton approved, but it is doubtful that if he opposed the repeal, that the veto could overcome the congressional majority.
(My main objection to Clinton was that he seemed to "go-with-the-flow" rather than leading.)

The Glass-Steagal Act already was being flouted, banking and securities were being combined. This was encouraged under Kennedy, and his Comptroller of Currency, James Saxon, who issued a regulation permitting national bank to have combined (comingled) accounts. The Supreme Court opposed.
In the 1960s and after, investors were withdrawing their funds from the banks and placing the money in the securities sector. The SEC allowed market funds to allow money market funds could redeem shares, and approached the functions of the commercial banks.
Furthermore, the savings and loan companies were essentially free of Glass-Steagal regulations. Gradually, FDIC-insured banks approached securities or investment banks.
In the 1970s, banks were allowed to authorize investors to make withdrarals to invest in securities. Banks were authorized to privately place "commercial paper."
President Reagan appointed regulators who favored deregulation. Bas were allowed to establish subsidiaries to deal in securities.
U.S. banks were having a hard time in the new marketplace, unable to compete internationally.
UK and Canada, and all others (except Japan), were combining activities and forging ahead.

Essentially, Glass-Steagal regulations were already dead before Clinton became president. The Gramm bill put fini to the banking-securities separation.

Putting the blame for 2008 on Clinton is a classic case of oversimplification.

Many complications, the development of hedge funds, and abuse of hedge funds, which seemingly allowed bankers to enter into loans whick were not backed, led to the crisis in 2008. That is almost another story.