John Kennedy actually said...

We are deeply convinced that the achievement of this objective would serve our best national interests and the national interests of all the...

It would make it easier to prevent the spread of nuclear weapons to countries not now possessing them.

A GUARANTEED end to all nuclear testing in all environments is a fundamental objective of the free world.

Such a treaty would result in a definite downward turn in the arms race as it is represented by testing to develop weapons technology.