I was trained as a Navy engineer, and believe that Engineering is one of the most underrated aspects of Star Trek (though Enterprise did a good job with some of Trip's story lines. "Similitude" is one of the better Trek episodes, IMO, both from an engineering and a dramatic standpoint).
-----------------------------------------------
As a preliminary step, I try to break down both engineering and science tests down into multiple parts rather than one lump roll and coordinate my technobabble with the engineer to explain what each roll represents and requires- to generate more of a "Trek feel" to the interchange.
In that regard, the Next Generation Technical Manual is an invaluable help.
The following links will prove useful as well:
http://www.ex-astris-scientia.org/
http://www.ditl.org/
http://en.memory-alpha.org/wiki/Portal:Main
I award circumstance bonuses to their rolls based on good roleplaying and good knowledge of Treknology, medicine, and/or science.
It almost always helps the story along if the players have at least a general idea what they're characters are trying to accomplish and how they're going about it.
Of course, I've been gifted with some genuinely brilliant players who enjoy that sort of thing.
Your mileage may vary.
-----------------------------------------------
When the crew wants to push the systems beyond specs or squeeze extra juice or speed out of the engines, I generally demand to know HOW they're going about it: increased deuterium flow, borrowing power from life support, etc.
For most purposes, I'll accept "running at maximum output" to take the ship to her top emergency speed, maximum power, etc.
When they truly wish to go above and beyond, however, I demand they get more specific.
The reality is that, no matter how brilliant your engineers are, the ship has only so much to give. Emphasizing that (without being too heavy handed) tends to give your players a feel for what they can demand of the ship, which in turn both "breathes life into the ship as a character" and tends to have a dampening effect on power gamers who insist that they should be able to push the ship to warp 13 if they just plug enough tricorder batteries into the EPS grid.
-----------------------------------------------
One of my favorite techniques is to allow the players to bypass the safeties (and/or running the reactor at higher pressures/temperatures) than it is normally rated for: then begin making reliability checks and making "Hmmmm" noises.
With high Target Numbers, the characters begin burn their Courage points to keep the successes rolling in- but as the number of points left in their reserves drop, their blood pressure rises. Of course, the more courage points they've invested, the greater the likelihood that a failure will be "catastrophic" (this is both a function of increasing target numbers and common sense).
Good story telling- and "subliminal" details that their characters will notice (warning lights, ominous groans and noises coming from the ship) help to set the tone.
If you are consistent in this technique, your players will get to know when the ship is simply "complaining"- and when the noises represent genuine danger.
This "over-charged reactor" technique was a bit over-used in The Hunt For Red October- mainly because there were no obvious consequences. ("Give me a hundred and ten percent on the reactor! We're already there? Then give me a hundred and fifteen!").
Used sparingly, the technique works well to ratchet up the tension and drama, but as Narrator, you need to remember that pushing machinery like that causes things to break- and the characters will be the ones picking up the pieces after even a minor failure.
In the novel (but not the movie), the Alpha Class Soviet submarine E.S. Politkovsky was lost because her reactor plant was pushed too hard for too long and failed catastrophically.
It is a truism in the Navy that safety regulations are written in blood, and that decisions have consequences.
If your players are routinely bypassing the safeties, pushing the equipment too hard, or demanding the impossible from their equipment, feel free to increase the TNs of repair and maintenance checks, and introduce equipment failures more commonly.
The recent collision at sea between the U.S.S. Essex (LHD-2) and the U.S.N.S. Yukon was directly attributable to the fact that Essex and her crew were routinely pushed well beyond what the Navy could rightfully demand of them- and her worn equipment failed at precisely the wrong moment.
Taken to extremes, your characters should realize that the safeties are there for a reason- and the big, red, flashing danger lights are exactly the color of human blood in zero gravity.
The first lesson should be (relatively) painless.
The second should be less so. (One of the nastiest things I did to a Chief Engineer was to trap him in a control booth- safe and secure- while a major EPS conduit breached and vaporized half his engineering crew just inches or feet away).
Voyager used a similar technique in "Year of Hell" when the doctor was forced to close a hatch on two crewmen trying to evacuate a deck about to breach.
The third lesson should be as harsh as circumstances demand.
-----------------------------------------------
The danger to these techniques is that detailed technical discussions (particularly those involving a fictitious vessel, dubious physics, and resolutions that are more often driven by drama than logic) tend to bog down the game.
Left unchecked, they can utterly destroy the pacing you need to create to keep the game entertaining and your players as a whole engaged.
Moreover, the Starfleet Judge Advocate General tends to take accident investigations very seriously- and inquiries and investigations can have a lasting impact on your stories and on character's career plans.
The fictitious captain of the E.S. Politkovsky died with his crew. The real-life Essex skipper- despite numerous exculpatory circumstances- is now flying a desk and staring at early retirement.
While it's necessary to keep your engineers engaged in the game (moreso than the occasional dice roll), you need to take care to strike the right balance for your group. Above all, though, you need to be consistent in what your crew can demand of their ship- or risk destroying the the suspension of disbelief.
Hope this monologue helps .