INFO and how come nobody is answering OP's question... specifically the "because the girl was rude" part?

The baggage claim is a no-nonsense part of the airport. It's hectic and full of people trying to shove through with their luggage. Your kid shouldn't even be there, let alone your drink, that obviously isn't in your hand because it was somewhere (where? on the belt? on the traffic barrier post?) where it could get knocked over.

Also, the woman never asked for compensation. She opened up with "your kid did that," and stood there like a busboy expecting a tip. I don't see the connect-the-dots process that ends with me taking my wallet out. She paid five dollars for a plain iced coffee (no way the mocha-frappa-whatever was five dollars in an airport), put it down somewhere stupid, and paid the price, and now she wants you to bail her out like it's her right.

If the conversation went down like "excuse me, I'm really sorry, and I shouldn't have left it there but I just bought that drink with my last five dollars and your child spilled it, would you mind hooking me up because I still have a three hour drive," and you told her to go for a quick jog, then I'd say yeah you were unnecessarily stingy, like you said, she was just rude.