You are supposed to use both Molex connectors on a PCI-e power adapter. Each wire is only rated to carry so much current, so the load is spread to two sets of wires. Being a low end graphics card, it's probably okay to use a single lead, but your power supply isn't a good match whether you use one lead from the adapter or both.
You're very close to the limits of that power supply. It has a single 12 V rail rated at 20 A, which in a great deal of cases is being generous by the manufacturer. Just off the top of my head, 18 A is actually the proposed standard (which a lot of supplies don't follow) for 12 V rails that are compliant, so it's entirely possible there could be safety cutoffs that are rated at 18 A but don't actually function until 20 A, hence the rating.
A GTS 450 graphics card can draw 10+ amps on the 12 V rail, depending on the loading, which along with your CPU (assuming a 65 W CPU [which is being generous]) and motherboard, hard drive(s), and any other peripherals drawing from the 12 V rail, may actually put you over the limit on the 12 V portion of the power supply.
If you never run the GTS 450 at full capacity, you might be okay, but it's not a high end card so it should be fairly easy to push the card toward it's maximum with any current gaming title. Only thing I can see preventing that would be an exceptionally weak CPU.
If you exceed the limits of your power supply, you can expect the system to spontaneously reboot, power off, have random BSOD errors, TDR errors with the graphics card, and the possible spontaneous self-destruction of the power supply. If the power supply fails, adding to the low cost nature of it, it likely doesn't include all manners of electrical isolation and protection, so you should anticipate it can damage other components in the process.
Being an old machine, it's probably not the biggest deal if it self-destructs. Maybe keep a fire extinguisher on hand, and be sure anybody using it has the good sense to be able to unplug it if something gets out of hand.