I've done this probably a dozen times at least. *LOL* (We used to fly a lot when we lived in Alaska.)
Only once did we have one damaged and it wasn't in a bag. Definitely take advantage of the free big plastic bag (if you don't use the box) and have them put it in there. Otherwise, it could get caught on any number of moving belts, etc, or if it's raining it may be wet when you get it back (I know it would dry eventually, but that would totally be a pain when you're traveling - BTDT).
A box would protect it really well I'm sure.
Oh, and if she sits in a sling or stroller, those are REALLY worth it to have with you, especially if it's a big airport.
Have a great trip!
