Kirk accosts her, guesses her name, and guesses wrong, proving that he doesn't know her. She gently corrects him. He doesn't know (or care) who she is, all he knows is that she's temporarily in his way. He doesn't see her as a sex object, he sees her as a speedbump. He's gentle about it, but the only attention he gives her is to gently pull her out of his way, since she seems to be too dim to understand that she should step aside.
Both she and Kirk demonstrate that they are not attuned to one another one bit. She's not interested in Kirk, which explains why she's not super-attentive to what he wants at the moment.
She maybe is distracted by her interest in Lt. Gary Mitchell. She goes for Mitchell and Mitchell goes for her right back. She and Mitchell hold hands as the Enterprise attempts to broach the barrier at the galactic rim, and this happens right in front of Kirk as if neither one thinks they have anything to hide.
Moments later, when Gary is felled, Kirk again has to chase Smith out of his way to show active concern for his fallen comrade. We see no more of her, so we learn nothing more about her. Mitchell quickly forgets her as soon as Dehner comes on to him.
Yeoman Smith is the romantic equivalent of a redshirt. Yes, minor female characters like Smith were sometimes casually thrown away, but male characters were thrown away by the bushelsful like autumn leaves, so it's not like the gals were especially picked on.
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