The presence of water under your sink strainer basket can indicate many things. You might have a bad p-trap, a split pipe, a broken seal around the sink or faucet, or it might just be the sink strainer. The sink strainer is quite simple to replace and you can buy a kit in a large box store or hardware store. You may have the entire kit or you may just have to replace the putty that seals the strainer in the sink and the lock nut. Release the drain tube under the sink that carries the water away.
Use the tweezers to undo the slide nuts that hold the drain tube in the plastic sink strainer basket. Move the drain tube out of the way so you can see. Set up a flashlight under the sink unless you have an assistant to hold it for you. Remove the prefilter basket. Use the adjustable wrench to loosen the coupling nut on the bottom of the filter.
You will probably have to put the tweezer handles in the two holes in the signature hole of the housing while putting pressure under the sink basket strainer sizes of the coupling nut. Take all the washers and the casing and basket out of the hole in the sink. Scrape the putty using the spatula. Clean the area thoroughly and make sure it is dry.