My rubber refers to the 2 big pieces at the bottom of the individual sinks as circled, it's probably dripping wet because of the sink's condensation beneath as well.
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These are sound-deadening pads - to absorb the sound when water hits the stainless steel sink. Mainly bitumen or tar-based with adhesive-backing to stick to the stainless steel surface, they should not be dripping wet as this will cause adhesive delamination.