What can cause a sink's pivot rod connector to leak water?
what might the cause be? 



The connector isn't tight, so the pivot assembly is loose, so water can work past the seals.
It was assembled with dirt in the seals
A seal has failed or is damaged
Something else.



  Can water still be reaching the pivot rod itself?


Yes.



I would disassemble, clean and inspect. replace any worn or damaged parts - especially rubber/plastic seals around or in the pivot assembly. Reassemble and test.



Example from Grohe:

  

8.4 45265 Nut and bushings