: Elrond is a half-elf, and it is he (implicitly) that Boromir is referring to, along with Gandalf and the other Elves - Boromir is endulging in a bit of Racism - implying only pure Dunedain would remain uncorrupted (Irnonically Elrond and Boromir are from the same stock - although separated by a few hundred generations).
: Half-Elves originate with the marriage of Luthien and Beren a man and an elf, also the case with Elwing and Earendil. Their offspring were allowed to choose which race they would belong to - the immortal elves or mortal men - Elrond chose immortality whilst his brother Elros chose to be mortal and founded the nation of Numenor. Elronds children were not true half-elves as they were the product of an elf and a half-elf, but were gievn Elven immortality until such time as Elrond left middle-earth at which time they must choose to go with him and become full elf or remain in middle-earth and become mortal - it appears that all his children chose to be mortal as Arwen marries Aragron (and their children are fully Mortal) and the boys Elladan and Elrohir remain in Rivendell - although some seem to think they still passed over sea later, personally I don't think they could have done.......
: And the Man from La Mancha is enchanted to find a familiar face on the Board.......
: Don Quixote

Was the choice for all half-elves, or just Eldarin? I seem to remember a mention of an Avarin (or Nandorin, I'm really not sure) half-elf that was the ancestor of the Princes of Dol Amroth. Would they have automatically become mortal, or have had a choice?