Maybe the non-used elevator decides to wind up being used. Maybe maintenance?

Fair suggestion, but I can assure you that the elevator isn't going anywhere ... they were in it when it crashed into the basement level earlier in the game! 

Is the actual climb important to the story, or is it merely important that they get to the top? I once made the mistake of playing the scenario included in the Space: 1889 rulebook which features a long, long climb. It was utterly tedious. Too many rolls, too difficult to introduce real tension and interest, too easy to scupper the game by seriously injuring the characters. Since then I've tended to look differently at similar obstacles.

With the fire in the basement and the rest of the plot on the fourth floor I would suggest that the important thing is not whether they make the climb (let them succeed, they ought to be competent enough: don't kill anyone off for something as trivial as getting to the next good bit) but what they might lose in succeeding. Assume that everyone gets to the fourth floor, but run a few quick rolls to see how well they managed it, encouraging the players to assist each other. A "failure" might mean that a character dropped some equipment (too late to go back down for it, the basement's on fire!) while a more seriously fluffed roll could be an injury, possibly in addition to losing gear or weapons. If another character turns back to help then they might avoid the injury but still suffer some equipment loss. It might be in their interest to split the gear up rather than risk everything in a big, heavy bag. On the other hand, good rolls indicate that they race up ahead of the flames in record time, scorched and grubby but essentially unhurt.

Lost gear doesn't have to be essential stuff, by the way. Leave the door open for players to improvise with other things, maybe just making their task at the end harder for having to use a few jerry-rigged parts. If climbing the lift shaft doesn't seem like an interesting thing to game to you, then I'd suggest hurrying along to the really interesting stuff and dealing with the climb in a more cursory way.

All interesting points, BigJackBrass, and I thank you for taking the time to respond. I think we are both on the same page insofar as -- "just move the story along" -- seems to be the path of least resistance!

In retrospect I could have named the post: "Ideas on making a climb interesting". I've done a fair bit of climbing (as a player  !) over my roleplaying career and each experience has seemed to be exercises in dice rolling, GM narrative and varying degrees of fiat.

I think I'll take the easy route. 

This is a late post to this thread, but elevator shafts can be very interesting locations for a climb. The shafts are also frequently used as a "right-of-way" for all sorts of other infrastructure, from data communications cables, to electrical cables, to plumbing. Some shafts have maintenance ladders to access to such utilities - making climbing easier. Of course some elevator shafts are only elevator shafts. How the characters approach the climb can add interest. Are they climbing up hand and foot holds along the walls or do the use the elevator cables? An elevator in a 4 story building is most likely a counter-weight design rather than hydraulic. Would they do something risky like cut the cable and ride the cable up as the counter weight falls? Depending upon portrayal of the condition of the building (like new vs old and poorly maintained) things like exposed wiring, or fragile plumbing can add excitement as a characters is nearly knocked loose the spray from a fitting on a water pipe that came loose when pulled on to lift themselves.

It all depends upon the atmosphere you want to create. if you want to give you player a respite, make the climb uneventful. If you want to increase a sense of danger and risk, use the elevator shaft climb to have more things go wrong and increase the sense of danger for the players. Events that cost players lost equipment (as mentioned above) are another great way to increase the sense of cost or sacrifice to achieve the goal of the mission with serious character injury.

Not too late a post, Paul, the game continues tomorrow evening.  Thinking about it, I'm planning on making the climb fairly straightforward -- your suggestion of pipes and wiring will actually aid them, as it offers something to hold onto. Basically, though, I think we will move to the "main event" in fairly short order.

The two- or three-part adventure they are on has already taken three game sessions, so I certainly don't think they will be missing out on anything if I get them to the next stage of the story quickly.