1 of 2 things happened. The hole you droped the sink into was a little to small for the sink and the caulk did not come into contact with the counter top or there was not enough caulk, bead of caulk was to small.
Let the old caulk cure and then remove it well, silicone caulk does not like to stick to it's self. Dry fit the sink and be sure the area of the counter top where the caulk will be conacts the sink bottom. The counter must be clean and dry apply 1/4" or so bead of caulk to the sink and carefuly install sink. When you look at the sink edge, before you install it, with the caulk on it you should be able to see that the caulk will touch the counter when installed.
The basket strainer has a rubber and paper washer they both go under the sink, rubber to the sink bottom then the paper(friction) washer next. The friction washer lets you tighten the large nut. Use plumbers putty between the top side of sink and basket strainer unless Mfr. or instalation instructions say not to. Read them.
Have fun !!!!!