Boromir held out long against this choice; but when it became plain that Frodo would follow Aragorn, wherever he went, he gave in. 'It is not the way of the Men of Minas Tirith to desert their friends at need,' he said, 'and you will need my strength, if ever you are to reach the Tindrock. To the tall isle I will go, but no further. There I shall turn to my home, alone if my help has not earned the reward of any companionship.'
Boromir's comments here show a curious sense of something akin to entitlement. He starts out promisingly; since he can't convince Aragorn/Frodo to come with him, he is willing to accompany them for a bit because it is "not the way" of his people to "desert their friends at need."
But he immediately develops the theme of "at need" by suggesting that the other eight cannot make it – even so far as the Tindrock – without him and his strength. (He is perhaps conveniently forgetting that it was Legolas who most recently dismayed the enemy by shooting a fell beast from the sky.)
This then takes yet a darker turn in his final sentence, that he will turn to his home at the tall isle "alone if my help has not earned the reward of any companionship." This is where the clear sense of entitlement comes in; his "help" (which he deems necessary to the others) should, in his view earn a reward. So he has, rhetorically, come quite a distance from it simply being "the way" of his people to stick by their friends when needed; he now thinks he deserves a reward for it.
The reward he names is "companionship"; the only proper way to show appreciation or gratitude for his help is to accompany him. Significantly, he does not, even now, suggest that any of the others have any help to offer him or Minas Tirith (although the Ring can never be far from his mind and if Frodo accompanies him, so does the Ring). Clearly, he believes he has the strength to reach Minas Tirith alone, but is setting up anyone who chooses to follow the original quest as a poor friend, ungrateful for all he has done for them, or worse.
It's darker still when you realize this whole little speech is meant almost entirely for Frodo's ears. It is clearly the four hobbits who have most needed the help of others to get as far as they have. But would Boromir be satisfied if Merry, Pippin and Sam rewarded him with their companionship to Minas Tirith, while Frodo went on with Aragorn, Gimli, and Legolas? I don't think so. Even if everyone but Frodo joined him, I think it would not be sufficient. The reward he wants is Frodo coming with him to Minas Tirith, thus bringing the Ring under the dominion of Denethor.