If we're talking about within the context of the original series itself, not a thing. It's about pitch perfect for the time and budget it was made in.
If we're talking about for doing a faithful reboot show or something along those lines, i'd still say almost nothing
in spirit needs to change. That being said, some of the things i'd probably do...
Exterior changes.
In general, i'd want to keep the changes to the design subtle. The objective would be to give the design more life. Somewhat in the tradition of Stan Winston when he went from The Terminator to Terminator 2. Don't change the design, improve the design. Give it more to do and more character. I'd take all the decals from the two pilot versions and the series version of the ship and combine them. Particularly some of the extra decals on The Cage version's saucer and bridge could be added to the series version just to give it a little more detail.
1. Windows with rooms, instead of just white lights. Have small people moving past them on occasion. That's an incredibly small detail, but I think it would increase the sense of realism greatly.
2. Probert's self illumination concept. I love the way the TMP Enterprise is lit up. I think that always gives the ship a better sense of life, and scale. This wouldn't change the design itself, but it would present it more dramatically.
3. Visible phasers, torpedos, and airlocks. Very delicately applied. I would probably hide the torpedo tubes and airlocks behind slightly off-color hatches. Similar to a sub marine. Very faint color variation. This would keep the design unchanged, until the appropriate dramatic moment. It would also give the battle sequences more drama as the "gun ports open."
4. Very subtle hull texturing. Just to add some additional color variation. Nothing extreme, though. Something that can catch some light here and there. Not turn it into an aztec'd nightmare or destroy the spirit of the design.
5. Different lighting states depending on flight mode and shipboard time. You could have fewer lights on when the ship is running the 'night shift' as an example. Impulse engines on or off, warp coils on or off, only when at warp would they be "on."
6. As far as the warp effect itself i'd do something similar to Mass Effect.
[yt]http://www.youtube.com/watch?v=l__hSYW_Qfc[/yt]
Having the visible light spectrum 'warped' around the ship with brilliant and beautiful color distortions thrusting past it. At a distance it might look like a shooting star.
Interiors
Overall i'd use a muted color scheme in the style of The Cage. I'd use a more dramatic lighting style, and i'd integrate more access panels and such in the corridors for maintenance and such. I'd also borrow more from the interiors of aircraft carriers and u-boats. I want to see more labels on things for one. Some rapid way to indicate which deck and what section you're in, too. (Could be as simple as Babylon 5's color bars.)
A lot of the control systems would be shifted from a 1960's futurism to a 21st century futurism. Touch screens/panels, holographic displays, some semblance of an actual user interface.
Also, for as crisp as the ships interiors look, i'd really want to make the Jefferies tubes look like crap. Lot's and lot's of exposed pipes and detail that is "behind the scenes."
The Bridge
Replace most of the hard buttons and switches with touch screens and interactive displays. Keeping things like Spock's scanner and Sulu's periscope, as well as weapon and helm controls tactile would be a good idea. Color wise i'd lean more towards "The Cage" as it looked far more realistic to my eye.
As others have also suggested, an alternative entrance and exit to the bridge would be prudent.
While it is good to keep the stations fairly multi-functional, i'd also want to see some very specific equipment to each station. Some sonar/radar/lidar type stuff for tactical and science. Indications of microscopes and different sensor systems. A stronger sense of just how the ship is maneuver from helm and navigation.
It would be really nice to see/show off the navigator actually plotting a course instead of pushing three buttons and saying "course plotted." Say, pulling up star charts in the computer and computation regarding the optimum plot.
The overheads over each station need to do a lot more too.
Also, i'd make the bridge view screen a window/heads up display view screen combo as it is in JJ's Trek. It really gave the bridge a sense of location. I know a lot of people were against that, but for me it just worked.
Transporter Room
Again, keeping the general Original series configuration, I would put the control system behind a protective barrier ala TMP, and i'd utilize the stasis feature that is hinted at subtly in the 2009 movie. I'd also have the back panels animated like they are in TWOK, just during transport.
Engine Room
Probably where i'd make the most dramatic changes. I've always felt the scarier the engine room is, the better. I want the matter/antimatter reaction chamber to be almost beastly. This is a power source beyond comprehension, and those that have to keep it in working order are true miracle workers. I'm not saying make it noisey with detail or make it a grunge factory, no. It can be an absolutely clean looking design, but it should exude a sense of INCREDIBLE power.
I love the TMP warp core for that very reason. I'd love to see that redone within a similar visual "language" of the TOS era. Or at least, combining elements from both.
Shuttle Bay
This set often get's overlooked in terms of what you could do with it. It's usually little more than a big open space where a shuttle lands.
Personally, i'd love to see it given more of an aircraft carrier treatment, with people in pressure suits "working the flight deck." People running fuel lines and operating maintenance equipment.
Maybe taking just a little bit from the "Aliens" playbook and have heavy lifter type machines to handle cargo. More indications of a "machine shop" for doing shuttle maintenance. Maybe add in some designated "cargo transport pads" as well.
Again, it's ultimately in my mind about augmenting what was there, and also about making that huge space serve as many functions as it logically can. Utilitarian, adaptable, and full of activity.