If you're playing the disc version, CSI: Miami was released in 2004 and made for Windows 98/ME/2000/XP so a problem saving when using Windows 7/8/10 is probably related to permissions.
The first thing to try is running as an Administrator -- right-click the game shortcut and choose "Run As Administrator" to start the game.
Has anyone come across this issue before
It's a very common problem on games made for Windows 98/ME/2000/early XP, and sometimes even later. I've seen it with Big Fish games made for Vista/7 too. The solution can vary with the game. Other things to try might be
- Compatibility mode to play
- Compatibility mode to install
- Installing somewhere other than C:\Program Files or C:\Program Files (x86)