Many feminists have dated or been in a relationship with someone who didn't quite understand feminism. While someone who's altogether anti-feminist may not be the best match for someone who is unapologetically feminist, people who are more agnostic can be open to learning about and understanding the movement — if you know how to explain feminism to your partner. It can be a frustrating process, and of course not every strategy is going to work for every relationship; since I've gone through it several times, though, I'll tell you what's worked for me.
First of all, know that it's not your duty to convert your partner into a feminist, make them understand something they haven't made an effort to learn (or simply don't want to learn) themselves, or put up with sexist behavior or comments from them. If someone isn't willing to change and you don't want to date them as they are, it may be better just to end the relationship. After all, if you're in a relationship with someone, it should be because you like them now — not because you like who they may one day be. And turning them into that person isn't your job.
At the same time, it's OK to do something that's not your job — if you want to. If your partner is open to discussion (and likely criticism), and it's worth the effort to you, here are some ways to help them understand feminism.