There were seats drawn on the blue print for the tower. Today there are no seats. Were they fitted and then taken out or were they never installed at all?
It is unfortunately far easier to prove the presence of seats than their absence. The newspapers worldwide covered the Eiffel Tower in great detail and I decided to check what they wrote about the elevators (which where obviously a particular point of interest). The National Library of New Zealand puts historical newspapers online and makes them searchable, that's what I looked at. One typical description comes from the Otago Daily Times, 14 December 1889:
Note how it mentions seats on the platform but not in the elevator. From Te Aroha News, 28 September 1889:
Numbers in these articles have to be taken with a grain of salt of course, I've seen articles published before the tower was opened talking about estimated numbers of 5000 people transported per hour which is clearly very far off. But seating 170 people would make an awfully large lift. And finally, Otago Witness, 24 October 1889:
Again, seats would be a good candidate to be mentioned, not only the wind-protectors. But this article doesn't talk about them.
Having decided from this indirect evidence that seats on the original elevators were unlikely I found the 2010 book Elevator Systems of the Eiffel Tower, 1889 (that's apparently the book where your illustration comes from). And judging from this book, your graphic refers to the elevator with the smallest capacity (in terms of persons transported), by Otis:
Oops! About the French elevators by Roux the book says:
And the two stage elevator mentioned in Te Aroha News was built by Edoux and had a capacity of 60 persons. The book doesn't tell whether people were seated in this elevator and the illustration sadly doesn't help because it shows the passengers while they are changing elevators midways.
However, the official guide of the Eiffel Tower (printed in 1893) says the following about this elevator:
And in fact, in this photo of this elevator you can clearly see that the women on the right are sitting. So to answer your question: yes, originally most people indeed took a seat in the Eiffel Tower elevators.