After Frodo is stabbed with a Morgul-blade on Weathertop, a shard of the blade remains inside the wound, slowly turning him into a wraith.
Aragorn knows that Frodo as a wraith would be enslaved under the power of the Witch King. It is only because of the unforeseen hardihood of Hobbits that he didn't succumb before reaching Rivendell.
Why didn't he take the Ring from Frodo (perhaps giving it to another Hobbit) in order to safeguard it?
There's no definitive explanation in the book, but several points seem clear.
First, what other Hobbit? If Frodo had proved too weak, would any of the others have appeared stronger? The options were two irresponsible kids and a servant. (While Sam did have much more strength than anyone -- including Sam -- expected, Aragorn can be forgiven if he failed to see that yet.)
Second, it seems clear that Gandalf had told Aragorn much about the task and about the Ring. (In The Prancing Pony, Aragorn is unsurprised when Frodo disappears and comments that he "put his finger in it".) Gandalf trusted Aragorn above any other Man and trusted him to guide Frodo in his place. He would not have kept Aragorn ignorant and risked Frodo and the Ring, so Aragorn probably knew most of what Gandalf knew or surmised about the Ring: How Frodo came to have it, the Ring's effects, Gollum, the Ringwraiths and all. And Gandalf would have told Aragorn, a friend of Bilbo's, about how hard it was for Bilbo to give it up.
Aragorn would thus have known enough to realize that taking the Ring from Frodo would have had serious consequences -- Frodo would certainly have resisted and the hobbits would almost certainly have sided with him -- and this would most likely have blown the party apart in the midst of wilderness and pretty much guaranteed the loss of the Ring.
Thirdly, Aragorn had to have been warned by Gandalf about the risk of taking the Ring himself. (Gandalf knew of the Ring's habit of making people lust for it.) Even aside from that, if Aragorn couldn't give it to a different Hobbit without blowing the company apart, he certainly couldn't take it for himself.
Fourthly, Aragorn had to have thought about all this and decided that as long as Frodo continued to resist, the Ring had best stay with him, but doubtless had a last-ditch contingency plan to take the Ring from him if Frodo fell to the wound of the Morgul blade.
Finally, he gave his word "I am Aragorn son of Arathorn; and if by life or death I can save you, I will", and he took that seriously.
Aragorn knew that the One Ring overmastered*, and some time after betrayed his ancestor Isildur to his death, so he was afraid of the Ring claiming himself as well should he take possession of it.
* The Ring overmastered Isildur at the very moment he was best positioned to destroy the Ring.
Aragorn could no more let Frodo keep the Ring nor make him give it up than Gandalf. From The Fellowship of the Ring, Book I, chapter 2, "The Shadow of the Past":
'All the same,' said Frodo, 'even if Bilbo could not kill Gollum, I wish he had not kept the Ring. I wish he had never found it, and that I had not got it! Why did you let me keep it? Why didn't you make me throw it away, or, or destroy it?'
'Let you? Make you?' said the wizard. 'Haven't you been listening to all that I have said? You are not thinking of what you are saying.'
...
Gandalf laughed grimly. 'You see? Already you too, Frodo, cannot easily let it go, nor will to damage it. And I could not "make" you except by force, which would break your mind'.
If Gandalf the Istar couldn't do it we shouldn't suppose Aragorn the Man could, even if they weren't out in the wilderness and Frodo weren't wounded.
