No! Caulking around the bottom will just keep it in and destroy the floor. The bottom is caulked to keep spills from getting under the toilet - not the other way around.
Since you say it's clear, clean water it isn't leaking from behind the caulk anyway. Anything that comes from BEHIND the caulk is most certainly "dirty".
I would reach around to the back of the toilet where the valves are and see if the connection or something at the valves is leaking.
Also, try to rock the toilet, Does it move? If so then it needs to be removed and have the seal replaced.
What ever the case, get it fixed quickly. Otherwise you are in for major rot and repairs.