I've shown calves of all ages ... even those that have been born at the show, on the truck, or at home the day before.
It certainly is a whole lot easier when they are 2 months +, but there are just a few precautions to take with the babies.
First, get a cage. Any sort of cage, that the calf will fit in. Secure the cage in the stalls next to mum, and put the calf in there. Dont tie it up, just leave it in the cage. Mum cant sit on it, and calf cant choke itself on the halter (I've seen that happen before).
Second, no-one gets to pet the calf. People are renowned for going round and patting all the baby calves, and shoving their fingers in the calves mouths, and that is how disease spreads round all the baby calves at the show.
Third, in the trailer or truck that you use to transport the calf, use a gate tied across a corner to make a small pen for the calf, again so he wont get squished by mum.
Fourth, when showing the cow/calf pair, just let calfie run after mum, and the handler of the calf walks behind the calf, pushing him on the bum if necessary. Dont try to walk out the front of the calf and lead him, because he flat out wont. Use a wheelbarrow if necessary.
Fifth, put together a 'Calf First Aid Kit'. In this I have antibiotics, needles, syringes, vet wrap, scourban etc.
Honestly, I've had baby calves get a touch sick (scours, runny nose) but I've treated them and never had any worse problems with calves dying from infections.
I HAVE lost two baby calves at shows (not ours, people I was grooming for), one was a couple days old and tied at the back of the stall divider, the cow sat on it (hence why I say get a cage). The other was a couple days old, and they tied it up with a halter in the truck, between the cows. When they got to the show (a few hours drive) the calf was dead, seemed like he'd spent the whole trip pulling back and choked himself (hence the calf pen in the truck/trailer).