I'm for bringing a stroller--but as compact as your family can tolerate. I don't like getting the strollers for the same reason I don't like changing my kids on changing tables...GERMS. If you are just looking for something for you kids to rest in--an umbrella stoller is perfect and doesn't take up to much traveling space.
Strollers are pretty safe at WDW--I have left diaperbags, blankets, etc. in them. The only thing I do not leave in them is Disney-ish items...so if I have the kids pooh bear blanket, a toy, or whatever, those go with us. We sometimes take the diaperbag with us and sometimes not...the kicker is if both kids are awake--or if one is sleeping. Hard to carry a kid and a bag.
Personalizing the stroller is best--it was a tip in the Disney with Kids unofficial guidebook. If someone realizes they are stealing something that belongs to someone...it is less tempting to steal. I would think the big clue is the stroller itself doesn't belong.
Alas, we have never had a problem with our stroller.
My only tip is if you have anyone has an infant seat--take it with you on the ride--if by chance it did get stolen, you lose your only car seat. Also if it rains, you wouldn't want baby to have a wet seat. Just my two cents.