The main quality of the star is that it is eternal. We can see this in the poet's use of the word "steadfast" and in his use of the actual word "eternal" to describe the way that the star watches over the world. This quality is very important for the main point of the poem.
The point of the poem, of course, is the idea that the speaker wants his love to be eternal and steadfast. Because he wants his love to be this way, he compares it to the star and he makes the eternal nature of the star (as opposed to its brightness) the major quality of the star that he emphasizes.
The qualities the speaker attributes to the star are steadfastness (line 1), patience (line 4), sleeplessness (line 4), and attentiveness (“gazing,” line 7). He assigns to the star the role of guard or watcher, thus suggesting that the star is like a divine presence, neither slumbering nor sleeping because of eternal guardianship over the children of God. The words “forever” and “ever,” combined with the attribution of the star’s being “steadfast,” provide a backdrop of permanence for the speaker’s love, which he wishes would remain forever firm and steady. Granted the comparative permanence of the stars, the choice of the star as subject is quite appropriate.