Which caulk around the sink in kitchen is a health-safe?
In general silicone is very safe for living organisms including humans. Cake molds, pacifiers and breast implants are made from silicone. 

To my knowledge every single sink installed in a table is installed using a silicone caulk. I have never seen a gasket for that purpose. 

Most carpenters and plumbers would use a silicone rated for wet rooms to install a sink. That means that they have some kind of anti-fungal agent in them. I don't know about the toxicity of such agents, unfortunately. However, do remember that mold is indeed unhealthy.