If you look at the Wikipedia article for DVI, especially the pinout, you'll see that DVI also can carry analog signals (the variant called "DVI-A"). That's why DVI to VGA adapters work: They take the the analog signals from the DVI-A pins, and route them to the corresponding VGA pins.
So if your graphics card doesn't support analog pins on the DVI connector (which it doesn't, it's a double link DVI port according to the specs, and in your picture it's missing the analog pins if you compare with the pinout on Wikipedia), then there's no way to convert it to VGA directly with a cable adapter.
There are other, more expensive adapters which are basically a small graphics cards which read the DVI/HDMI signal, and then produces a completely different VGA signal from that.
So use on of those, buy a different graphics card or use your old graphics card in addition to the new one, or buy a new monitor.
DVI and HDMI are basically the same digital signal. The main differences are that DVI has the optional analog pins, and HDMI provides for a copy-protection scheme (HDCP) that's necessary for some consumer electronics, but not an issue between computer and monitor. HDMI to DVI cable converters work fine both directions.
I would say the DVI to VGA; it will be cheaper. If you are willing to spend a few extra bucks then you can find DVI to VGA cables online; these should be cheaper than HDMI or DisplayPort to VGA adapters. https://www.amazon.com/HDE-DVI-DVI-I-male-Cable/dp/B002IWP87S
– MonkeyZeus – 2017-02-03T13:43:09.8477
@MonkeyZeus That's a nice thought, but OP is not going to be able to plug that cable into their GPU. The port on the GPU is clearly DVI-D (look at the flat slot on the right of the port in the OP's image, and note the lack of 4 more openings surrounding it). That cable you linked to is DVI-I, which requires those 4 pins, which are used to carry analog data.
– 8bittree – 2017-02-03T15:10:01.377@8bittree Yep. You're right. This whole fiasco has made me more knowledgeable. I read a lot about it today and know the difference between each modes now. – asprin – 2017-02-03T16:21:20.077
@8bittree hmm, I didn't look at that cable close enough. That cable in specific will not work with that GPU for sure. Unfortunately full DVI-D to VGA cables do not seem to exist and only adapters are available. Looks like DVI-D to VGA adapters are just as expensive as the HDMI and DisplayPort counterparts. So in this case I change my suggestion and you should go get an adapter for the DisplayPort because it will have actual latches on the male end and will be less likely to just slip out of the slot due to movement, vibration, or whatever. – MonkeyZeus – 2017-02-03T17:05:01.133
1NVIDIA video cards used to ship with a DVI-VGA adapter, but I haven't seen either an adapter or a monitor with only a VGA port in many years. Your monitor is probably long overdue for replacement. – Michael Hampton – 2017-02-03T20:31:18.357
@MichaelHampton The popularity of ports is probably dependent on the country. I just bought a brand new monitor a couple of weeks ago and it only comes with a VGA port. If it's also any surprise, TV's here are also still sold with coaxial cable ports. – JoL – 2017-02-03T22:26:55.130
Monitors have inputs ... – 287352 – 2017-02-03T23:57:53.483
Proper passive adaptors don't exist. Or when in doubt, just get the same model. – Journeyman Geek – 2017-02-04T08:20:53.367