If your stirrups are wet or covered with mud or shedding sand you might not want to run them up and get that all over the flap of your pricey French saddle.
There are also some types of stirrups that do not run up, and running stirrups up on calfskin leathers is rough on them.
I generally do run stirrups up, but honestly, as a safety issue it is a pretty minor one. What exactly are the stirrups going to catch on as you walk your horse from the groomed arena down the wide barn aisle to the crossties? If there is stuff sticking out that the stirrups could catch on, that’s more of an issue than the stirrups hanging down. If your horse gets loose and is running around, the stirrups getting caught on something are the least of your worries. Plus, in the rare event that it DID catch on something, most likely the leather would slide off the stirrup bar of the saddle. If you have a mouthy horse, sure you know you need to run up or cross those stirrups. But most horses aren’t reaching around biting at the stirrups.
People do hundreds of much worse things all the time. If the people you ride/board/train with are such that this is the worst safety infraction you can find to fuss over, I’d say you are in good shape. I would recommend NOT criticizing other riders who choose not to run their stirrups up–it’s not a serious enough safety issue. I DO get after people for tying horses in an unsafe/improper manner, letting strange horses get too close, riding too close to others, etc.
IMO, whether or not you run your stirrups up (or cross them) is more of an indication of how “well brought up” you are. If you were brought up in a proper barn, you were taught this as a matter of procedure/discipline and likely have retained the habit.