I don't get a successful connection between the computer and the RS232 port.
Actually you have not provided evidence of this assumption. More likely you are assuming that you have not established a good serial connection to the remote device and its RS232 port. You have not mentioned what RS232 device you are trying to connect. A serial port really doesn't do much without another serial port on the remote end.
How can I determine whether or not this adapter cable is functioning as it should?
If you have a voltmeter (e.g DMM), then you could check the TxD (on pin 3 of the 9-pin connector) is "marking" at about -5 to -15 volts. Ground is on pin 5.
The simplest method would involve first having a working RS232 connection, with a serial COM port on the PC, a "good" serial cable, and a responsive remote device. Substituting the PC's serial port with the USB adapter should preserve functionality, but on another COM port.
Establishing a working RS232 connection is not a plug-n-play operation. You need to obtain the serial port parameters of the remote device, and setup the PC side for the exact same baud rate, number of bits for character length, number of stop bits, type of parity, and method of flow control. Get any one of these wrong/mismatched, and you may get absolutely no data transfers or maybe garbage data.
You also need a proper serial cable. There is no one serial cable that will work in all circumstances. The PC RS232 port will behave as a DTE, data terminal equipment. Typically remote serial devices act as DCE, but that is not guaranteed. Obtaining the serial port pinout for the remote device is always advisable when you cannot make the connection work.
Edit: I once purchased five similar USB-to-RS232 adapters (actually they're converters not adapter cables), and one of them turned out to be a dud out of the box. I don't remember the failure symptoms. So it's possible that you may have a dud also, but you've provided absolutely zero diagnostic information or how you're trying to use or configure this adapter.
I've used USB-RS232 adapters with Prolific or FTDI converter chips. No problems with ten FTDI adapters. One Prolific adapter was DOA. The 5 adapters that use Prolific chips have a RS232 signal ground that floats => Chassis ground of the PC has to be tied to chassis ground of the remote device for a reliable link. The StarTech you have probably uses the Prolific converter IC.