In the book, Aragorn has Anduril in his scabbard, but it is broken, as he shows Sam. How could he defend himself in the wild? Is there any reference from LOTR or HoME?
Throughout the Lord of the Rings, willpower and lineage are far more powerful forces than mere arms. The primary weapons of the Nazgul are not their swords (dangerous though they are) but their voices. Sauron doesn't ever actually do anything in LotR, save command legions of followers and cause his enemies to quail in fear (mostly indirectly via the Nazgul). The Dead strike their enemies with madness, not with weapons. The siege at Helm's Deep was broken by the dawn and the appearance of Gandalf the White, not because the orcs were outmatched by Erkenbrand's newly-arrived forces. Although brute force does play some role (in destroying Isengard, in taking down the gate at Gondor, in the destruction of the balrog, etc.), it is usually secondary.
So I think it is consistent for Aragorn to routinely travel without any weapon at all, or with only a broken weapon, except in times of war. He commands respect--from Barliman, for example--even when being dismissed as a queer ranger, and from animals (the horse Hasufel in Rohan, for instance). In the wild, his stature as the heir of Gondor, and his long years of travel as a ranger, are probably enough (given that he exists in Middle Earth).
He carries the broken sword as an heirloom of his House not as a weapon. And just because his sword is broken it doesn't necessarily mean he has no other weapons. He probably came to Bree from Rivendell and was travelling light as the lands aren't particularly dangerous for a Ranger and he plans on returning before his next journey. He also knows weapons are no use against The Nine anyway.
"There is food in the wild," Strider said, "berry, root, and herb; and I have some skill as a hunter at need. You need not be afraid of starving before winter comes. But gathering and catching food is long and weary work, and we need haste." The Fellowship of the Ring
Remember that Aragorn has journeyed long in the wilderness and is able to procure his own food. Hunting requires at least a good-sized utility knife for skinning, cutting rope for trap-making, and a ton of other things. No good hunter/tracker would journey without a decent knife. So we know that Aragorn at least has one of those, maybe several, that he could defend himself with at a pinch.
I also think hunting would require perhaps a small bow/arrow, not for formal archery but close-range shots. Indeed, the LOTR movies portray Aragorn as having a small bow and a hunting knife as well, which you can see Viggo Mortensen sharpening during some scenes.
It always did seem odd to me that Aragorn should carry around the shards of Narsil, a precious and ancient relic of his house, on his long journeys in the wild. What if he lost it? What if it was stolen? And it clearly would be useless as a sword.
I speculate that he only sometimes carried it around on short journeys, and not all the time. He probably carried a proper sword at other times if he felt he would need it. Just because he had Narsil on him at the time he met up with Frodo doesn't mean he had carried it around always for his whole life.
Is it possible that it was broken but still usuable?
Could it have just had the top 6 inches broken off or something? That would mean it is still a functional sword and also is the sword that is broken?
He has two swords, Narsil isn't reforged until later in the series, before then he uses his other sword. An example of him using the other sword was during the fight with the Uruk-hai next to the river at the end of movie/book one. Narsil resides in Rivendell and isn't delivered to him until later in the series after it is reforged into Anduril.