Rinoa's outfit has no pockets or pouches. Therefore, she has nowhere to keep a handkerchief. I think it was quite decent of her to go outside to pick her nose. That way, there was less chance of her being seen, and she could simply flick it off the balcony rather than leaving it stuck on a table or floor somewhere. Of course, if Squall hadn't crassly interrupted her in his usual abrupt manner, then she wouldn't have had to pretend that she was brushing her hair away from her face. That's the worst part about it - she was trying to be courteous and discreet, but thanks to Squall's lunkheaded impertinence, she wound up having to wipe snot through her hair. Rinoa takes a lot of pride in her well-groomed appearance, so that must've been a real hassle for her.
Hopefully her nasal discharge blended in with those cute gold highlights, so it wouldn't be so visible.

That's why I think it's good that Squall and Rinoa got together: Squall would never pick his nose in public, but then he's so private that it's rare for him even to talk in public. Rinoa, on the other hand, knows that necessity is a powerful force. "When you gotta go, you gotta go". She's pragmatic yet also thoughtful - she proves this when she's forced to pick her nose, but chooses to go outside and keep it away from everyone else. She can't just walk around with a blocked-up head, but she's not going to upset the people around her by mining the glistening ore right where they can see her. Imagine what Squall would've done - he'd have held it in till he started to get sick. When someone asked if he was OK, he'd likely just sneeze snot all over them. Squall doesn't care about communication that much, so he wouldn't mind being so blocked up that he can't talk properly. Rinoa, however, is a true 'people person'. She'll do whatever it takes to ensure she can always help others to be happy, even if it means taking drastic measures - whether it's fighting against her father's army, or sneaking out the back to pick her nose. Same principle, different context. Squall would do well to learn from this.