I just recently noticed a vibrating water pipe under the drywall
ceiling of my partially finished basement. It vibrates when the first
floor toilet is flushed and the kitchen sink turns off.
Is there a way I can minimize the vibration
without cutting open the cieling?

You could try replacing the valve inside the toilet. But that won't help
the kitchen sink.

You may need to install water hammer arrestors at those two fixtures. These
absorb the pressure waves when you suddenly turn off the water.

Anthony