Blunt needles are a thing. They are used for example in electronics as a way to dispense tiny amounts of solder paste or flux on components pads on a PCB. Presumably when transferring solutions from container to container, you don't need a sharp needle, unless the container is closed with a membrane that needs to be pierced.

There are also gloves that protect against puncture. If you're using both sharp needles and non-protecting gloves then it is probably time to revise the procedures.