Yes, you can see through to the ground. I don't have a fear of heights so can't comment on whether it would be manageable for someone who does as that person might start off ok, but then as they climb higher get more uncomfortable. If you aren't able to get tickets for the Eiffel Tower elevator, perhaps just enjoy the view from the ground and have an alternate plan to climb something easier where you can get a view of Paris (and one that actually includes a view of the Eiffel Tower), like the Arc de Triomphe, or take the elevator ride up to the observation level at Tour Montparnasse, or the rooftop terrace at Printemps department store.
While the view from the top of the Eiffel Tower is nice, frankly, I thought looking up at the tower from the lawns of the Champ de Mars in front of it, or Trocadero behind it, or standing directly underneath it and looking straight up overhead, was actually more interesting and made for better photos/video during the day and at night when the light show happens.
EDIT: I'll clarify my answer better - yes, like Wil said there are kick plates on the stairs that you step on, but you are able to see through the metal pieces on the sides of the steps and that is the part that might make someone with a fear of heights a little queasy which is why they might prefer the elevator ride that is quick and painless.