But if NASA does this too much, won't Jupiter spiral into the sun?
However, that would require a lot of mass. NASA would basically need to build enough Jupiter-slingshotting spaceships for their combined mass to be a significant fraction of Jupiter's. This would probably involve dismantling entire planets or moons to build that much spaceship mass.
Yes, or about the mass of Jupiter, which is where the number came from--the article's comparison of the two.
[edit: added emoticon]