Can't say I have ever tried rock climbing but I can already see a disadvantage that you need a mountain to climb or nearby indoor rock climbing resort. Boxing is also without a shadow of doubt cheaper, some places even let you do it for free.
Boxing can be done pretty much anywhere, I train myself and my favorite thing to do on a nice day with a girl is take her to the park and teach her some moves on the focus pads. Not only is this a very cheap date but she will get hot and sweaty so you can offer her a shower at your house which is only 5 minutes away
But like anything else don't do it to pickup chicks, do it for yourself the physical and mental conditioning you get from boxing or any martial art for that matter is priceless. There is something about being able to handle yourself that gives you an aurora of confidence, you don't have to be Muhammad Ali to get this either.
Honestly give it a try, you won't regret it.