I'm a big fan of submarine dramas, which for hostility of surrounding environment -- and the necessity of devoting every square inch of space to multiply-redundant systems that keep you alive -- are a pretty useful analogue for space travel. If you're in a ship built for deep-space operations, cramped quarters should never not be a major element of the ship sets, because no matter how much your technology advances, you're still going to want to expend the maximum possible amount of power on your drives, shielding and environmental systems. Comfy living quarters will always be a second-tier priority.
Really committing to that sense of things provides an ever-present plot mechanism, because in a really hostile medium, the smallest mistake or malfunction can kill you, your resupply and refuelling concerns are constant (and can drive any number of stories), the sense of exploration itself as a dangerous... enterprise where your greatest enemy is your environment comes to the forefront.
Put the crew in compression suits instead of slacks and shirts for their uniforms. In case of a hull breach, they can just throw on gloves and a helmet and they're good to go. Same goes for away missions plus a nifty cowboy belt for their phaser. Uniforms would incorporate nifty stuff like smart materials to assist movement and regulate temperature, electronically tinted visor, HUD, and a tricorder in the gauntlet. Maybe a low-power forcefield like the life-support belts from TAS. Not enough to make them phaser-tanking Iron Man expy, but enough to protect against radiation and thermal hazards. This whole kit would reflect the precariousness of the crew's situation - they're out in deep waters on the strangest of tides, and they can't really afford to let their hair down too much. Something could go wrong at any minute.
The Enterprise is cramped, like a submarine, but clean and brightly lit, with design DNA that kind of makes it look like a high-tech version of a module from the ISS, looking more scientific than militaristic. Everyone has their own "sleeping cabin", but it's basically like one of those Japanese capsule hotels. Yes, even the captain's cabin. It's not cosy, strictly speaking, but would have a charm all its own.
Little things like that. It's mostly window dressing, from a storytelling standpoint, but anything that makes Trek try something new, in terms of futurism and general production design, is welcome in my book. ST09/STID isn't really that radical of a reinvention when you get down to it.
Shoot, do a Trekified submarine movie/series synthesized of elements from those films above. You don't have to have all that danger/heroics stuff every episode. Deep space exploration in a sardine can, meeting alien races who range from primitives to advanced technology. It would be like a modern sub discovering Tahiti or the Chinese 500 years ago.