Finishing a basement properly is tricky and standard building practices are ill-suited to the job. The trapped condensation in the walls you’ve got is common, it’s just that most people don’t see it because their walls are covered.
The short answer to your question is no, the condensation probably won’t stop when drywall is installed. It may be reduced, but probably not eliminated. And unfortunately, this situation often leads to hidden mold growth within wall cavities. It’s the main reason why so many basements smell musty a few years after finishing.
The main issue is typically caused by moisture vapour that migrates in through masonry walls. This is the source of the moisture that’s condensing on the inside face of your vapour barrier right now. Unfortunately, fiber-based batt insulation has no resistance to permeation by moisture vapour. It also happens to be a prime breeding ground for microbes whenever it gets damp.
All this is why the ideal way to finish a basement is to coat the bare masonry walls with an effective waterproofing compound, then use closed cell foam insulation on the walls. The compound stops inward migration of moisture vapour, and the foam is impervious to any moisture that happens to get past the waterproofing.
I’m afraid there are only a few options at this stage. You can forget you’ve ever seen condensation, install drywall and hope for the best. I don’t recommend this, but it is the same position that the vast majority of finished basement owners are in (even if they don’t realize it). Another option is to take everything down, then proceed from square one with a moisture-proof coating followed by foam-based insulation. You could also remove the vapour barrier and fiber insulation only, coat the walls between and behind the studs to moisture proof them, then have spray foam applied between the studs.