I think it's worth pointing out that - no matter if we are wishing or wanting - the expression "be mine" can easily sound creepy, possessive, or insecure. It's okay on a valentine (because we don't interpret everything in a valentine as literal fact), and it might be okay in a song (because we sing lyrics akin to poetry, not how we'd talk at the lunch table). If I were to approach a woman and say, "I want you to be mine," there's a decent chance I'd be answered with a restraining order instead of a date.
If we're talking about "appropriate" conversation, I'd have to recommend:
I'd like to go out with you sometime.
I'd like to get to know you better.
before I'd use either of the two options listed in the question.
That said, if I had to choose between the two verbs in the context of a budding relationship, I think want sounds more demanding than wish. Wish conveys a spirit of wistfulness and pining; I'd say it's close in meaning with:
I'm longing for you to be mine.
Want in this context sounds more primal or perhaps immature - like a child about to have a tantrum if he doesn't get his way. I want you to be mine might work in the context of a movie script, just before a steamy lovemaking scene, but I'd strong caution anyone against using the word "want" in the context of revealing emotional desires in casual conversation, especially for the first time with someone of the opposite sex.