Boromir and Frodo are the two members of the Fellowship who
sacrifice their lives on the Quest. Boromir sacrifices his life to be
the catalyst that propels the members of the Fellowship on the
different directions that they need to take in order to fulfill their
roles, whether physically or emotionally.
Boromir’s sacrifice of his life in view of Merry and Pippin both gives them an example of the nature of nobility and valor, and gives them the final appreciation that the Quest is a serious matter. It also demonstrates to them that they are valued in and of themselves. It’s been pretty clear to that point that Frodo is extremely valued as the Ringbearer, and Sam is valued as Frodo’s man (hobbit). But they would never have expected this incredibly valiant warrior who (absent Aragorn) would be the next ruler of Gondor, to value their lives so highly that he would sacrifice his own in an effort to protect them. I think it is a huge event in their maturity – their lives have been dearly bought, they owe debts which later lead them to take hasty yet crucial and effective actions. It also demonstrates another reason why Gondor is worth saving, worth dying for, if it is the home of people of such valor and generosity.
Boromir’s death scene in the movie is much more effective for me, so I base the next bit on that. Aragorn has been wracked with self-doubt and indecision about whether he should pursue his destiny to become the King of Gondor. During their travels together he and Boromir have been able to observe each other, fight with each other against the Enemy’s forces and exchange thoughts on matters that are close to their hearts. Their respect, esteem and affection for each other has grown. Boromir has been closer to the actual rule of Gondor, having been raised as the son of the Steward; he has also been closer to the actual forces of Mordor. He would be a necessary ally for Aragorn, but he also has knowledge and experience that would make his judgment more meaningful to Aragorn, who would supplant him. As he dies Boromir extracts an oath from Aragorn that he will not let their people fail, nor the White City fall; he also gives him an oath, that he would have followed Aragorn and accepted him as his king. The respect and faith of this most valiant warrior are not given lightly, and Aragorn knows that. In the same way that his death shows Merry and Pip that they are valued, Boromir’s oath affirms to Aragorn that he is worthy of his destiny, his doom. And now Aragorn is bound to seek it by virtue of his oath to the dying man. He still has doubts to overcome, but he can no longer waver or dither about whether he is worthy or strong enough for the throne. Like Frodo, he must take it or die in the attempt. He would not have come to that decision in time to take action if it were not for Boromir.
Of course Boromir is the catalyst for Frodo’s decision to finally leave the Fellowship behind and seek Mount Doom alone. He is the first to succumb to the Ring’s power and his attack on Frodo has a powerful impact on the poor hobbit, both from the physical threat perspective and from the grief of placing the members of the Fellowship at risk of suffering similar fates if they remain in proximity of the Ring. Later, Boromir is the catalyst for Faramir’s judgment that Frodo may be trusted, both in the veracity of his tale of how he and the Fellowship parted ways, and that he must be allowed to continue his Quest since he is the only one who has been able to withstand the power of the Ring.
By not being able to protect Merry and Pip from the Orcs and the Uruk Hai, Boromir is the catalyst that sends Gimli and Legolas on the hunt with Aragorn in pursuit of the prisoners. Alliances are made which lead to the defeat of Isengard and the distraction of Sauron from Frodo and Sam. Boromir sacrifices his life to truly begin the Quest, while Frodo gives his life to complete the Quest by bearing the Ring as it grows in power and achieving its destruction.