Route (MacBook Air) audio output through (Dell) LCD Monitor speakers
Okay so a couple of things are at play here. First of all, DVI doesn't carry audio - so even though the Mac's Thunderbolt port is capable of sending audio, when you convert it to DVI that audio stream gets lost. This can get confusing though, because it is possible to send a HDMI signal over a DVI-D cable, which will carry audio. But that's technically not a DVI stream any more. 


  A DVI-D source can output HDMI signals including audio (e.g. ATI 3000->series and NVIDIA GTX 200-series).[9] Some multimedia displays use a DVI >to HDMI adapter to input the HDMI signal with audio. Exact capabilities vary by video card specifications.


Source: Digital Visual Interface article on Wikipedia

The next thing is whether or not that port is an Input or Output. I've got an LG monitor down the end of my desk that has a 3.5mm Jack on the back, but it's an output. Meaning if you plug in a source that supports audio (in its particular case, HDMI) then you can pass that audio out to a set of desktop speakers. But it doesn't have any built in.

If it is indeed an input, you just need a 3.5mm to 3.5mm audio cable, and connect the headphone output of your MacBook to the speaker input on the monitor. 

Something like this would do the trick: 3.5mm Stereo Jack Cable on Amazon

The upshot being, you can't do audio and video over DVI connection alone.