Caulk around kitchen sink faucet?

Caulk around kitchen sink faucet? My faucet looks like this: Should I caulk around it or not? I am not talking about the sink itself and the backsplash ... as may be seen from the picture, I (re-)caulked these already. If I look under the sink where the faucet is it looks a bit spongy (but dry, not wet), not sure if this is water damage or not. Also, the sink starts to wobble a bit from time to time which drives me nuts. I always re-tighten the screws underneath but issue re-appears. I think it might be due to water/moisture seeping through the tiny gap. If caulking: How? If no caulking: How is this supposed to work?