I'm with you...the whole "I'd be down" and some of the other phrasing sounds completely corny and borderline douchey. Handing a girl your number then explaining that you'd be down with hanging out is cringe worthy.
Also, I don't know why so many of you think him giving her his number, unsolicited, is a good play...it's not.
How about you go to the cafe regularly, start some witty banter, make small talk, feel her out/determine if there's a little bit of chemistry (at least superficially), and then ASK for her number. Giving your number is a bad idea for a bunch of reasons, the big two being that 1.) it's "safe" and rejection proof (thus showing a lack of confidence) and her accepting your number in no way guarantees she'll call. 2.) even if she's interested or at least intrigued by you, she may not call out of her own fear or insecurity. Sure, plenty of girls will call a guy, but a lot of gals like being pursued or at least sold on the idea that you're worth their time. Giving your number is a gamble, and doesn't really send a self-assured, confident message.
So, scrap this give her your number in a cute clever way...that's bad teen movie romcom bullshit, and step up to the plate AFTER you've already established something of a friendly relationship. Ask her name, ask her a few casual, not too personal or nosy questions. Especially if you're new to the area, size her up and ask her questions about the area, things to do, places to go, blah blah...if you're charming, funny, and engaging, you're probably in unless she's gots a man. At the end of a particularly fun convo whilst she's brewing your joe and the cafe isn't busy, just tell her how much you enjoy chatting with her and that you'd love to see her outside of the cafe...whoever said to ask her to coffee.....bahahaha...she's a fucking barista, the last thing she needs is a coffee date....