How do Hamlet and Polonius affect the outcome of the downfall of Denmark in Shakespeare's Hamlet?

In short, Denmark does not “fall” in Shakespeare’s Hamlet.  In fact, Denmark still exists today.  (In Hamlet, it is the continued life of Horatio that secures Denmark's ability to thrive further.)  However, the King of Denmark, Claudius, certainly does fall.  Perhaps this is what you mean by “the downfall of Denmark.”  Both Polonius and Hamlet play a part in Claudius’ downfall, but Hamlet plays a much bigger part than Polonius. 

Polonius plays an...

In short, Denmark does not “fall” in Shakespeare’s Hamlet.  In fact, Denmark still exists today.  (In Hamlet, it is the continued life of Horatio that secures Denmark's ability to thrive further.)  However, the King of Denmark, Claudius, certainly does fall.  Perhaps this is what you mean by “the downfall of Denmark.”  Both Polonius and Hamlet play a part in Claudius’ downfall, but Hamlet plays a much bigger part than Polonius. 

Polonius plays an indirect role in Claudius’ murder while Hamlet plays a direct role in that murder.  Polonius is simply the fool of the play while Hamlet is the protagonist.  This relates directly to their involvement in the downfall of Claudius.  Polonius does agree to spy on Hamlet for Claudius.  This really is not a good idea and leads to Polonius’ death behind “the arras” due to Hamlet mistaking Polonius for Claudius.  Hamlet, however, directly affects Claudius’ downfall.  Hamlet does spend the entire play stalling and/or over-thinking the matter.  However, Hamlet (finally) kills Claudius at the end of the play with the same poisoned rapier that Laertes used on Hamlet.  One could also say that the poisoned cup (meant for Hamlet) also results in Claudius’ death.  Of course, it is Hamlet that forces Claudius to drink from the cup he meant for Hamlet to drink.  In this way, Polonius indirectly leads to Claudius’ downfall while Hamlet directly leads to Claudius’ downfall.