Not a definitive answer, but a quick bit of googling suggests a few things. Mainly, in short, you can't, not with a standard port of any sort from a standard PC without a suitable printer thats part of a POS system.
Firstly, that these drawers connected to printers in some manner, and the pinout depended on the printer (see this and this, and without a suitable printer, this may not be doable at all.
However, apparently they are fairly simple devices - the printer would trigger a solenoid that would open it, its device independant other than the cable, as such either a standard interface, or something very simple, and these use 12V. You might be able to get the drawer connected to a RS232/Serial connector if you have one, since the voltage levels appear to be within spec but some testing might be needed. If this fails
there's no simple way to get a 12 V signal out of a system, so you would likely have to work out how to trigger the device, possibly using voltage off the power supply of the system, some kind of relay, and possibly a microcontroller - the exact process would be far out of the scope for SU.
I'd recommend finding the pin out on the cash drawer side, and proceeding from there.