Let's think cause and effect for a moment. Let's suppose you somehow are on the Enterprise with foreknowledge of what will happen. Let's say you even manage to persuade Kirk of this. So Kirk does something to prevent Ilia from getting beamed away.
Did anyone stop and wonder what happens then? V'ger doesn't have their mouthpiece on the Enterprise to negotiate with. It's very possible they decide a ship full of "carbon units" is more trouble than it's worth and just purge it. Especially after they terminated their means of contacting them by stopping them from absorbing Ilia.
So by saving Ilia, you don't save her for long, and you doom the whole planet Earth. Congratulations.