If a person goes up (e.g., on a staircase), then человек поднимается. If an elevator is going up, then лифт поднимается. That's the same verb for the person and for the elevator. However, if a person is coming down then человек спускается while if an elevator comes down then лифт опускается. I'm surprised that one direction has the same verb while the other direction uses different verbs. If the language can handle the idea that going up is the same thing for people and for elevators, why doesn't it do that for going down? Is there a commonly used verb for going down that can be applied to both people and elevators?
It is not true that the verb "опуститься" is used only for objects and "спуститься" is used for people.
There is a tendency to use the verb "спуститься" to denote a deliberate action and the verb "опуститься" for an uncontrolled movement:
Птицы спустились с неба. - The birds flew down to the ground.
Воздушный шар плавно опускался на землю. - The balloon was slowly going down.
However, in my opinion, there is also another difference, which may be even more important to consider, when you choose, which verb to say. "Спуститься" denotes movement down relative to an object (a tree, a mountain), while "опуститься" refers to downward movement in general. In particular, the verb "спуститься" is more likely to be used with preposition "с" - "from" and "в\на" - "to", than the verb "опуститься":
Вася, спустись с дерева! - Vasya, get down the tree!
Я спустился в подвал (по лестнице). - I went (downstairs) to the basement.
Сорочка лёгкая спустилась c её прелестного плеча. - The light chemise fell from her lovely shoulder.
Паром спустился по течению реки. - The ferry came down the river.
Этот лифт спустился с пятого этажа. - This elevator came down from the 5th floor.
Лифт наконец-то опустился, и мы вошли в него. - The elevator has finally arrived (from above), and we entered it. (actually, you could also use "спуститься", if you cared enough about the fact that it has been above before)
Занавес опустился, и зазвучали аплодисменты. - The curtain fell down, and there was applause. (we don't care where the curtain fell from)
На гору опустилось облако. - A cloud covered a mountain. (literally: fell down onto the mountain)
So, to answer your question: you can use both verbs for people and for elevators, it does not depend that much on who or what came down, it rather depends whether you care from where it came down.
Language is not math. And even strictly scientifically: going up and going down are two very different things - it is not like going left and going right.
I agree with Olga's answer. I could have really put my answer as a comment to hers, just putting it as a separate answer because of the size.
We are talking here only about "спускаться" and "опускаться" in the meaning of going down. They have other meanings as Lyth points out.
"Опускаться" accents the moment of final smooth breaking before to stop on the ground. It is especially noticeable in the form "Он опустился на землю" - here most of the people imagine the moment of landing and smooth breaking before it (притормаживание). It may mean the whole process but just accents the ending part.
"Cпускаться" - accents the process of moving legs, hands and whatever else during the descending. "Он спустился на землю" - here people will imagine the end of possibly long process of steps of descending.
Because there is no natural moment of soft breaking when a person or object goes up there is no special word "оподняться" for moving up. Even when there is some ceiling, usually objects just hit it when go up - there is no smooth breaking before it. OK, the smooth breaking may happen but the language did not need a special word for it because of rareness and insignificance of this phenomenon.
Just for illustration of non-equivalence of up and down movements, let's look at moving through the water for objects. In Russian there are two more or less equivalent (except for the direction) verbs "тонуть" and "всплывать". In English there is "to sink" for "тонуть" but for "всплывать" there is no direct equivalent (or I could not find it :). There is "to surface" but it accents the moment of emerging on the surface - not the whole movement, and there is "to float" which mostly means (according to Webster) "to rest on the surface, or be suspended in liquid", or even moving in any direction: "yellow leaves floated down" (= плыть in Russian).
I think it's ok to use спускается for an elevator. Спускаться bears meaning of "to go down by own will" and when used with inanimate objects indirectly animates them (which is perfectly fine).
Опускаться is more difficult; the difference between спускать and опускать is like "to bring down" and "to bring down gently/steadily". This extra detail plays a dirty trick on reflexive suffix "-ся" and gives an extra meaning of "being brought down (by someone or something)". More than that, опускать has a second meaning "to omit" (опускаться — "to be omitted"), and a third — "to bring down" in a moral point of view (опускаться — "to fall").
As an alternative you may use "идти вниз" (literally "go down") — "лифт идёт вниз", "человек идёт вниз" are both normal phrases.