What could have been done to stabilize the Tacoma Narrows bridge?

The Tacoma Narrows Bridge was built in a location with high winds. High velocity winds can create an aerodynamic instability called flutter. Flutter may be mitigated by a stiff structure. That bridge was very flexible in torsion (more commonly known as twist). There were many warning signs of a disaster - the bridge had wild oscillations in high winds many times before it collapsed.

If you ever see a video of a stop sign rotating about its post in a hurricane, as often done by on scene news crews, the phenomenon is completely analogous. Of course, during "normal" winds, the stop sign would not flutter - during the hurricane the wind speed exceeds the speed when we see the onset of flutter.

All aircraft are designed to ensure the maximum speed of the aircraft is below the speed at the onset of flutter for its wings and other lifting surfaces. Here as well, the wing design may be stiffened if required to raise the flutter onset speed.

After the Tacoma Narrows Bridge collapsed, many bridge designs were reviewed by engineers to ensure they would not have the same fate. The Whitestone Bridge, which crosses the East River from Queens to the Bronx in New York, was one that was stiffened by side bracing to raise the torsional stiffness and remove the concern about flutter. To my knowledge, that bridge was never subject to winds at a high enough speed in its pre-stiffened state to induce severe oscillations like those seen on the Tacoma Narrows Bridge.