GN commuters seem to have a lot more of a problem with 387s than their counterparts who travel on Gatwick Express, experiencing near enough identical trains, just with a different paint job! And when it comes to the Southern and GX Electrostar fleet, those trains typically attract far fewer air con complaints than anything running north of the Thames. I really don't know why this should be.
GTR has actually spent a copious amount of money and time in air con overhauls for older trains, and the HVAC systems on 700s and other newer stock do seem to be getting better. I spent several exceedingly uncomfortable journeys on freezing 700s last winter, but I'm yet to have a problem a year later. The same applied with air con seeming to get better towards the later part of summer. As for 387s, I must admit I've never really found any with major problems.
One thing is certain, though - trying to open the windows in the event that the HVAC has not failed is likely to make the problem much, much worse. Those windows do very little and are chiefly for emergency ventilation and to mitigate against total air con failure in the summer - but opening them for "a bit of air" in the middle of winter is likely to direct a freezing cold jet of air at the air handling systems, which will not do them any good at all.