It's always a good idea to remove ANY parts you can before flying with a stroller, including the hood, the napper bar/tray, the parent tray, and yes, the cupholder. Put them in your checked baggage. As a word of advice, don't fill the basket of the stroller while going through the airport--you will have to fold it twice, for security and prior to boarding. If you can't manage to carry your stuff (carseats excepted) and your child onto the plane at the same time, you have too much stuff. Backpacks make the best diaper bags for travel, BTW. A waist strap is useful; so that it doesn't slide over your head when you bend down to tie a shoe, etc.
The most common way for breakage to happen is if the stroller comes unfolded during handling. If this happens, the airline employees will NOT take the time to figure out how to fold it; they will simply cram it shut however they can. Be sure that your stroller is folded and tightly secured before leaving it on the jetway; if the latch comes undone easily or if it has a button-fold, putting some kind of strap around it to keep it folded is very wise.
The hold of an airliner (even the "small hold" if the plane has one) is usually a very greasy place. I always put a ripstop sleeve around the upholstered parts of my stroller before I checked it; even bungeeing on a large trash bag with the bottom cut open would help.
It is crucial to make sure that the gate-check tag is secured in a spot where it is very visible and cannot just slide off (especially important w/ umbrella-handle models.) If the gate-check tag is not readily visible to the handlers, they may send it to the carousels.
