It's certainly true that she was First Lady, and it's not a fact irrelevant to her political career (she certainly was an active political figure in the '90s). But more recently she served in the U.S. Senate and as Secretary of State, and narrowly lost the Democratic nomination for President. That's the record she's running on, and it's the record that (most of) her opponents are running against. Any one of those facts would make a more appropriate introduction.
Now this random story about New Hampshire is hardly worthy of comment. But some of these things accumulate. They're not things you can simply “cut out” like online trolling, they're widespread habits that no single person can stop. They're as much the result of sexism as the cause. All of the people running for President and covering the race have been shaped by decades of sexist double-standards. While the sexism of any particular act during campaign season can be hard to separate from the constant stream of often-unfair attacks by and towards everyone, looking at the big picture, you can see its mark.
ADDENDUM: As this post was inspired by the journalistic convention of introductions... I feel like pointing out that I added in Clinton's first name at her first mention while editing the post, just because I was thinking about the convention, even though I thought it didn't match the style of the first paragraph, which I might describe as, “awkward informal references to assumed shared context, because I'm too lazy to write a good introduction or even link to a relevant article, because this whole post started out as a Facebook status update before it got too long, and I was WAY to lazy to edit it down.” I just noticed that I didn't add Sanders' first name at his first mention. I'm going to leave it like this, because, damn, this post is a trainwreck of writing whose victims are every form and style of the English language. Like my dad said when my brother was making fun of his receding hairline, "Would you make fun of me more if I went on Rogaine or got a toupée?" And the answer was, of course he would have made fun of him more.