So, in our last session, one of my players decided to ask out an NPC (Eliza Bloom, if you're familiar with her). This completely stumped me, as I have no idea how to handle it. On a pure Social Skill level, she has him absolutely thrashed; something like 3-12, can't remember off the top of my head. But he's being persistent by sending her flowers.
Any thoughts on how I should proceed? I have no problem with a relationship developing, but I want to proceed in a fair and reasonable way.
Before we go into how I deal with romance, ask yourself if you even want this to happen at all. Because romances can be problematic, and they can even undermine your adventure if you're not careful. If you say no, it doesn't matter how well he rolls on Negotiation or Con or whatever, he fails. Period. If an NPC is powerful enough that being seduced by the PC can give him extraordinary powers, just don't allow it.
Generally speaking, if an NPC isn't particularly important, I just reduce things to a die roll or two. I don't get too far involved. Many of my players will pick up someone at a club and take her home and forget about her in a week. Of course, I get to roll to see how well protection worked and if they get something nasty, or she comes down pregnant.

Edit: However, if you decide to go through with this, and the aforementioned NPC has the PC blown away in the socials department, you have a number of options:
1. Is she out of his league?
By League, I mean, does she have more money than he does? It might also refer to how much class she has, and if she does, if she associates with certain high level people in the area, then she may not be able to afford even being seen with the PC, no matter how classy he may be. If he can't afford to keep her in a Lifestyle she's accustomed to living in, she may treat him like a boy-toy and nothing more. The odds that any real genuine romantic feelings may develop in her for the PC is astronomical.
2. If she's in his league, or lower, she may still not like him.
I never let Con or Negotiation rolls dictate if a person is in love with anyone. All the skill rolls do is help to validate those emotions. However, if, as you say, she will thrash him in the social department, she may be a manipulative bitch who will play him like a fiddle. Women see men, first and foremost, as a means of money; sorry, but it's true. Sure, a lot of women are nice and won't try to waste our money with frivolity and can't be bought into having sex, but a lot of women are, and Emily has an option of milking the PC for every spare nuyen if she really wants. Play this very carefully; just have her toy with him, egging him on to buy her gifts and other things, and over time, she'll have him paying rent. The key, though, is make him earn any affections from her at any given moment. That's if you want Emily to be a real bad girl.
3. She falls in love with him.
Maybe Cupid rigs the game, and she falls in love with the PC. Now you have the option of the PC benefiting from this relationship.
You don't really have to go into any details. You can keep everything topical, just saying "You and Emily have a nice evening on the town and spend the night with each other." And that's it. That can all be code for "They had sex." If your PC deliberately says he's not having sex, at least not until marriage, okay, but if he doesn't, assume that he does.
Relationships can be clutter for your campaign, so be sparing about them. You shouldn't let them interfere too much in your sessions. They are complicated, even if they're storybook relationships where strangely nothing bad ever happens, and it's paradise for both her and him.
When I used to run D&D 3.5, I had a PC fall in love with an NPC who was a noblewoman. And I had made the mistake of dwelling too much on this, because the PC wanted to become a knight of the realm in order to be able to wed her. Well, that took up a good portion of every session, even when the PCs were off adventuring to some forsaken dungeon, and the others got real tired of it, fast. It almost ended my campaign abruptly. So I had to keep things short and topical, even to the point that they didn't interfere with the session.