This an interesting question, because the poem alludes to two different cultural conceptions of heroism -- the early, pre-Christian, Germanic conception, and the Christian one.
The seafarer himself exhibits some aspects of the pre-Christian heroism: He's tough and voluntarily takes on dangerous, harsh challenges. But he's not the classic hero from the pre-Christian age. He doesn't slay monsters or do battle as a warrior. He doesn't receive the rewards of treasure and praise. Instead, his acts are heroic in a different sense. He doesn't work to glorify himself, but to obey the will of God.
In the second half of the poem, he draws analogies between his physical voyages on the sea and his attempts to live up to the demands of Christianity. He, a good Christian, is a voluntary exile for God. Thus, he employs some of the Germanic heroic virtues -- being tough, choosing a life of danger -- in the service of doing God's work.
We can see this unfold in the text. First, the poem describes the narrator as having faced great dangers on the sea. He has toiled alone, endured " earfoðhwile," or times of struggle, coping with terrible, tossing waves that sometimes threatened to dash his ship near the cliffs (lines 1-8). He's been covered -- hung about -- with icicles ("bihongen hrimgicelum" in the original text) and showered by hail (line 17).
Of course, suffering and endurance aren't necessarily the same thing as heroism. But the narrator also tells us that he chooses to face these hardships. Whereas city dwellers enjoy an easy life on land, he feels the pull to return to the sea (lines 33-38, from the Owl Eyes translation on eNotes):
And how my heart
Would begin to beat, knowing once more
The salt waves tossing and the towering sea!
The time for journeys would come and my soul
Called me eagerly out, sent me over
The horizon, seeking foreigners' homes.
So the seafarer is tough, brave, bold, and ready to seek out danger. But in the middle of the poem, the narrator notes that traditional Germanic heroism is becoming a thing of the past, and in any case the rewards it brings -- riches, praise, and a glorious reputation on earth -- don't last forever. And while the narrator might seem to be nostalgic for those times, he hints that traditional Germanic heroism is egotistical. In the second half of the poem, the seafarer likens his physical adventures on the sea with the self-abnegation and asceticism demanded of the good Christian, and he lays out how the Christian's virtues differ from the traditional values of Germanic heroism (lines 106-112):
Death leaps at the fools who forget their God.
He who lives humbly has angels from Heaven
To carry him courage and strength and belief.
A man must conquer pride, not kill it,
Be firm with his fellows, chaste for himself,
Treat all the world as the world deserves,
With love or with hate but never with harm…
Thus, the poem is outlining a different conception of heroism, one that de-emphasizes personal glory and requires turning the other cheek. It's consistent with what scholar Catherine Woeber sees as the central theme of Christian heroism in many Anglo Saxon poems -- something that means
"more than brave battling on the side of God; rather, it is complete submission to the will of God."
Marsden R. 2004. The Cambridge Old English Reader. Cambridge University Press.
Weber, Catherine. 1995. Heroism in three Old English poems: A Christian approach Koers 60(3) 359-379.