A MapBasic application is bitless. That means is neither 32 bit nor 64 bit.
You can run most MapBasic application in 32 bit as well as 64 bit versions of MapInfo Pro.
But there are limitations to this. Your MapBasic application might rely on external resources, for example a C++ or Delphi DLL. If this DLL is 32 bit, you can't launch the tool in 64 bit version of MapInfo Pro.
Also 64 bit MapInfo Pro have retired a few MapBasic commands, such as the Graph and Crystal Report support. These commands will not work in 64 bit. Similar 64 bit have added numerous new commands that will prevent you wrong using the MapBasic application in an older version of MapInfo Pro.
Specifically to the RegEx support. This is based on a 32 bit DLL which makes it impossible to use this tool in 64 bit MapInfo Pro.
You can find a similar discussion on the PB Knowledge Community which also holds a solution for 64 bit: Will anyone upgrade MapBasic Regex to work in 64-bit