First of all, check up the software, what port (COM1 or COM2, ...) it uses. Try to select another port it the settings. If it's impossible, check up quantity of ports in system. If your new computer has built-in video and only one COM port on the back panel, it doesn't mean the system board has only one COM port. Can be, that the additional connector for the second port is not installed. And check your computer's BIOS settings - the COM ports have to be enabled.
If problem persists, you can use old serial mouse (if you have it) to check up the COM port. Try to connect it to COM port (and disconnect PS/2 mouse) and look, is it found whether or not. If found - port works.
You can also use external modem to check out the COM port, if you have it.
There are special programs for ports checking, but I don't use them. More simple means usually suffice. Can be, such programs someone will advise another.