Just take out the old caulk and replace it with a good silicone caulk...one for bathrooms. Yes, you should have it to avoid the accidental leaks getting under the commode and into your floor. FYI: the smell could be from the floor itself and you need to replace the wax ring under the commode and check the subfloor. :o/
If there is an odor, its probably leaking and needs new wax seals. Caulk doesnt smell on its own.
new caulk can have a strong (chemical) odor... especially a silicone for several days (is this what you smell?). I have back and forth on caulking a toilet to the floor... this should be only an aesthetic decision and absolutely not necessary for the control of water or a sewage smell. if you smell sewage or see water. think @Tile By Gregory is dead on. replace wax ring...
I have heard to NEVER caulk around the base of a toilet. If there is a leak under the toilet the caulk would hide the problem, and the subfloor could be ruined.
that's a good point @Anna ... and I agree that if a sewagy smell is the problem here. caulk is not the answer. cheers.
I agree with Anna Parker and Tile by Gregory. As a matter of fact, one of the contestants on All American Handyman was criticized for caulking the base of the toilet. Mike Holmes explained that caulk will hide a leaky wax ring and allow the subfloor to rot. If there is a bad smell, the wax ring is probably leaking and needs to be replaced.