Forcefields in Trek are good at exerting a force. While this would be good for keeping air from escaping, the field might equally well simply keep the wearer pressurized to one atmosphere without involving any air: the forcefield would be a skinsuit providing compression while in vacuum, and armoring against external pressure while underwater. The air supply would only ever operate around the mouth of the wearer.
Although instead of a belt, a necklace would then probably be the wearing mode of choice...
Circulating a bit of air across the body, by a peristaltic effect of the field, would serve thermal management needs. The air that gets into Kirk's mouth and nose would then have a good dose of his body odors, but that's what you get in the average spacesuit, too. Getting fresh air in and used air out, instead of mixing the two, might be more easily and accurately done with peristaltic forcefields than with helmets, really.
We get bona fide leaks in TNG era spacesuits, so I'm more willing to believe in air tanks than in rebreathers. But the heroes might have James Bond style miniature air bottles pressurized to 1,500 atmospheres or something, and suitably cooled to boot.
Touching is probably best done with the forcefield fully between the skin and the object - otherwise the heroes in this episode would get rather horrible frostbite! But it must feel awfully slippery, walking on forcefield soles like that...
As for TOS VFX, I trust they could have done a brief flash of light when the field is turned on, after which it would go invisible like every good TOS forcefield. Would that look cheesy? Well, cheesy is what you always get with spacewalks, no matter what your fictional hardware. Utterly futuristic invisible spacesuits might in fact have worked the best!
Timo Saloniemi
Now I've just the extras to see and then it's vamoosh!!! 
And the Romulans even less so...
