You're probably thinking of (and slightly misremembering) "Once Upon a Time" by Nadine Gordimer.
Points that match:
- story around two pages long
- paranoid parents
- child killed in barbed wire
Points that don't match:
- the child seems to be older than a "baby"
- the parents aren't necessarily afraid of someone harming their child, just paranoid in general: to steal a phrase from Spagirl, they're being very Daily Mail in their fears.
Relevant quotes from near the beginning and end of the story:
It was not possible to insure the house, the swimming pool or the car against riot damage. There were riots, but these were outside the city, where people of another color were quartered. These people were not allowed into the suburb except as reliable housemaids and gardeners, so there was nothing to fear, the husband told the wife. Yet she was afraid that some day such people might come up the street and tear off the plaque YOU HAVE BEEN WARNED and open the gates and stream in... Nonsense, my dear, said the husband, there are police and soldiers and tear-gas and guns to keep them away. But to please her - for he loved her very much and buses were being burned, cars stoned, and schoolchildren shot by the police in those quarters out of sight and hearing of the suburb - he had electronically controlled gates fitted. Anyone who pulled off the sign YOU HAVE BEEN WARNED and tried to open the gates would have to announce his intentions by pressing a button and speaking into a receiver relayed to the house. The little boy was fascinated by the device and used it as a walkie-talkie in cops and robbers play with his small friends.
One evening, the mother read the little boy to sleep with a fairy story from the book the wise old witch had given him at Christmas. Next day he pretended to be the Prince who braves the terrible thicket of thorns to enter the palace and kiss the Sleeping Beauty back to life: he dragged a ladder to the wall, the shining coiled tunnel was just wide enough for his little body to creep in, and with the first fixing of its razor-teeth in his knees and hands and head he screamed and struggled deeper into its tangle.
Found by searching the web for
short story baby protection barbed wire. The barbed wire detail is (obviously) very distinctive and memorable.