We know Galadriel tempted the members of the Fellowship, did she tempt Boromir with visions of taking the Ring and saving Gondor?
In the books, Boromir willingly accepts the judgement of the council that the Ring should be destroyed, but after the meeting with Celeborn and Galadriel his personality seems to change.
Was the temptation offered by Galadriel in some way responsible for Boromir's fall?
No. Boromir had been tempted by the Ring ever since he first saw it in Rivendell. Sam, for one, saw this clearly, as he explains to Faramir at Henneth Annûn:
[Said Sam]. 'It strikes me that folk takes their peril with them into Lórien, and finds it there because they've brought it. (...) Now Boro—' He stopped and went red in the face.
'Yes? Now Boromir you would say?' said Faramir. 'What would you say? He took his peril with him?'
'Yes sir, begging your pardon, and a fine man as your brother was, if I may say so. But you've been warm on the scent all along. Now I watched Boromir and listened to him, from Rivendell all down the road – looking after my master, as you'll understand, and not meaning any harm to Boromir – and it's my opinion that in Lórien he first saw clearly what I guessed sooner: what he wanted. From the moment he first saw it he wanted the Enemy's Ring!'
The Two Towers, chapter 5, "The Window on the West"
I believe it is likely Galadriel tempted him with the ring, and in doing so re-ignited a pre-existing idea to take the ring, but to be clear lets break this down into three parts.
Boromir pleas for the ring to go to Minas Tirith, to help Gondor in its defense against Mordor.
“ Why should we not think that the Great Ring has come into our hands to serve us in the very hour of need? Wielding it the Free Lords of the Free may surely defeat the Enemy. That is what he most fears, I deem.”
However after it's decided that the ring will go to Mordor, Boromir seems to be content with the decision.
“So be it,’ he said. ‘Then in Gondor we must trust to such weapons as we have. And at the least, while the Wise ones guard this Ring, we will fight on.”
After setting off Boromir pays no special attention to Frodo, playing his part in the Fellowship, carrying faggots of wood to keep everyone alive, and carrying Hobbits through the snow. It is not until after talking to Galadriel that he makes mention of the Ring again.
“To me it seemed exceedingly strange,’ said Boromir. ‘Maybe it was only a test, and she thought to read our thoughts for her own good purpose; but almost I should have said that she was tempting us, and offering what she pretended to have the power to give. It need not be said that I refused to listen. The Men of Minas Tirith are true to their word.’ But what he thought that the Lady had offered him Boromir did not tell. And as for Frodo, he would not speak, though Boromir pressed him with questions. ‘She held you long in her gaze, Ring-bearer,’ he said. ‘Yes,’ said Frodo; ‘but whatever came into my mind then I will keep there.’ ‘Well, have a care!’ said Boromir. ‘I do not feel too sure of this Elvish Lady and her purposes.”
It would seem that Boromir had been tempted, and quickly went into denial of the fact. He seemed to be trying to solve what Galadriel was doing, and may have been tempted by the Ring, saying pretended to have the power to give. Boromir is fully aware she could not give the Ring as she pleases and this may be what he's suggesting. Boromir also suddenly has a great interest in Frodo, and what she said to him while staring at him.
It is undeniable that this is a turning point for Boromir, where his temptation for the Ring grows tremendously. This is supported by his obsession with the Ring as they paddled down the Anduin. But whether this is due to Galadriel's temptation or not is unclear, although it is quite likely.
Merry and Pippin in the middle boat were ill at ease, for Boromir sat muttering to himself, sometimes biting his nails, as if some restlessness or doubt consumed him, sometimes seizing a paddle and driving the boat close behind Aragorn’s. Then Pippin, who sat in the bow looking back, caught a queer gleam in his eye, as he peered forward gazing at Frodo.
It seems clear from the final chapters of the First Book that Boromir has become obsessed with Frodo, which I now believe is likely due to Galadriel's temptation.
I haven't read the book but when the Fellowship enter the Lothlórien and come up to meet Celeborn and Galadriel, she enters Borormir's mind, making him realize that his beloved people are in great danger. That vision makes Boromir reconsider the power he could possibly acquire if the ring was in his possession (and then how he could save his people from Mordor). He later confesses to Aragorn the vision he had and his fear regarding the future of Gondor.
Galadriel knows how humans are attracted by the power of the ring and could have cast this vision inside Boromir's mind in order to push him to try to steal the ring from Frodo when they were alone, making Frodo realize that the only way he could achieve his quest was on his own (or at least with someone he trust more than everything like Sam) without the threat of humans' avidity for power (the same thing reoccurs later with Faramir). I think Galadriel was trying to help Frodo rather than the Fellowship itself and that she already planned Boromir's decadence, possibly even his future sacrifice.
