[Boromir88]

Elu brings up some good points. Aragorn, by letting Frodo go would only be protecting him.

Now Tuor is right in saying that Aragorn contemplated following Frodo (I believe along with Sam and Gimli), but he still thought in unwise to send everyone with Frodo and that they needed to remain hidden.

There is no doubt Aragorn would have been a good asset to Frodo and Sam. With Aragorn in their company, there would be no reason for Gollum. Aragorn had tracked down Gollum all the way into the Dead Marshes, and I think he knew the terrain well enough to get them to Mordor. So, there would be no point for Gollum, which comes the big question, who would the Ring tempt?

The ring likes to set it's eyes on one person (though as Elu says the movies show it as tempting everyone within the proximity it seems). But, the Ring likes to go after one person. It first had Boromir, an easy target for the ring, and the Ring had no need to go after anyone else. Boromir was easy prey for it. When Boromir was gone, Gollum steps in, another easy prey and target for the Ring to go after. So, if we have Aragorn, Frodo, and Sam, who would the Ring go after and tempt? My guess would most likely be Aragorn, as there's no reason to tempt Frodo and Sam has proven to be too simple to what the Ring has to offer.

Though Aragorn already had a chance in Bree to take the Ring, being in Bree amongst a crowd of people and 4 Hobbits is much different then marching into Mordor with no one around and 2 Hobbits.

Also, I stick to my original answer in that Aragorn had other duties, he had another path that seperated from Frodo's. Aragorn had a different journey he had to take. His path was to be Gondor's King and strike up a war to distract Sauron from Frodo and Sam. Aragorn I doubt would be able to do this if he goes with Frodo and Sam. Plus, for the fact again this is Isildur's heir, and for sure would catch Sauron's attention if he spots Isildur's heir walking into Mordor. He finds the Hobbit's, but only sees them as harmless spies. So, it would only be a matter of time before he would spot Aragorn with them, and no doubt he wouldn't have seen Isildur's heir as a harmless spy.