Nobody move! What's that? You were already just sitting there? Well, that's not a very energetic crowd. In this poem, the speaker seems attracted to the dynamic energy of the swans. He draws a sharp contrast between their seemingly infinite vigor, and his own lack of vitality. A good way to get that idea across—of aging, fatigue, and decline—is with images of stillness. It's not just the speaker who's sitting on his duff. Look closely, and you'll notice that this idea is carried out in a other ways, too.
Lines 3-4: "The water / Mirrors a still sky." So which is it? Is the water still? Is it the sky? Likely it's both. There just isn't much going on in the setting.
Lines 19 & 24: In both of these lines "still" is used to mean "continues," as in "You're still watching that Discovery special on tree slugs?!" However, Yeats is a poet, a person who made a living with words. Are you telling us that he couldn't figure out another way to get this idea across without repeating himself? Well, we admit that it's possible. But, we'd also like to suggest that the repetition of "still" here—even when used in another sense of the word—is meant to underscore the speaker's lack of energy.
Line 25: This is perhaps the most intriguing use of stillness in the poem. The speaker's just finished propping up these swans for being mad energetic in the previous stanza. They've been paddling and flying and basically trying out for the swan Olympics. Now, though, they're back on still water, just "drift[ing]." What's up with that? It's as though these swans aren't actually real, observed animals. Rather, they tend to act in a way that suits the speaker's mood—energetic when he admires them, still when he contemplates how they'll leave him for good.