I'd like to suggest that all men interested in approaching women in public start carrying their own contact info on a slip of paper or business card to hand out. If it's a cold approach, I envision the conversation going like this:

"Hi! My name is John. I'm sorry if this seems forward, but I think you're [cute][beautiful] and if you're single, I'd love to get to know you. Would it be ok if I gave you my number?"

Ask this from 6 feet away and only approach closer if she says yes. If she takes it, say, "great! Well, I'm gonna get back to [grocery shopping], but I look forward to hearing from you." This gives her two opportunities to say no (no to the number, and then no to actually calling even if she takes it), and doesn't create any safety risks in giving out her own information. Some women still aren't going to like it, but if you wait for 100% agreement, you'll be waiting forever, and I can't think of anything more broadly appealing than that.