Frodo had to go (alone if must be, but I think it's best that Sam would go along with him) to have hopes of destroying the Ring. There are a few reasons why:
First, Aragorn's task was different then Frodo's. Frodo was going to destroy the Ring. Aragorn's duty was to reclaim the throne of Gondor. Also, Gandalf was using him (and aid from the Elves, Rohan...etc) to strike up a war with Sauron to keep Sauron focused on them and not on Frodo. The whole point of the war (and sadly Legolas is right in this) was a diversion, to distract Sauron from finding out about Frodo and Sam. So, Aragorn's path, to reclaim the throne and help Gandalf strike up this war was different then Frodo's.
Secondly, Frodo would be put in great danger if Aragorn went with him. Aragorn is the heir of Isildur, and surely having the heir of Isildur march into Mordor would cause much more focus on Frodo and the Ring, instead of two little hobbits that are seen as "mere spies." But, having Aragorn follow Frodo into Mordor would be far too dangerous for Frodo, because sure this would attract Sauron's attention. When Aragorn uses the Palantir Sauron already fears that Aragorn has the Ring. It would be far too disastrous to have Aragorn go with Frodo, just like if Gandalf was to go all the way to Mount Doom with Frodo.
The main weapon the Free Peoples had against Sauron was secrecy, they had to keep their true purpose hidden, which was to destroy the Ring. Aragorn's path to reclaim the throne was different, also, it wouldn't be so secret anymore if Sauron spots Isildur's heir waltzing into Mordor. Hopefully, that helps, welcome to the downs.
