Dalton is alive, no doubt about it. He was pretty weak when fought in Zeal but by the time of the Dimensional Vortices he is truly a force to be reckoned with. He became even more powerful than Magus, who was once slaughtering Guardia with just his minions. It would be pretty anticlimatic if he died off-screen after conquering Guardia and (possibly) killing Crono and Marle. Bottom line is, we need answers.
I wouldn't say 'no doubt about it' in this situation. All we have is doubt right now. I think listing what facts we actually know from both games about the rise of Porre and the Downfall of Guardia might help with these questions.
Facts we know:
(1) Dalton, through a dimensional abnormality, arrives in 1000AD. He aids/manupulates Porre into a war with Guardia.
(2) The Kingdom of Guardia is defeated and razed by Porre. The Masamune is lost during the battle.
(3) The Masamune is not retained by Porre and is returned to the Denadoro Mountains (while there is evidence for their placement in Divine Dragon Falls it seems more likely given quotes about the mainland that it returned to Denadoro Mountains).
(4) Porre is still a major power in the world during Chrono Cross.
(5) Guardia does rebound at some point and rebuild, retaining it's sovereign status though it's size and military strength are unknown.
Things we don't know:
(1) The status of Dalton. Whether he survived the war, still runs Porre with an iron fist, or opened a small bake shop in Medina (mmmmm cake). We know nothing about this man. It would be anticlimactic for him to have died, but the suspicious lack of information about him in Chrono Cross is strange considering he was never a subtle man at any point we had seen him.
(2) The status of Crono and Marle. As much as pro-Dalton fans might brush them aside, we do not know what role they played during the fall at all. They might be living, dead, or unliving. I feel it is likely that one of them regained the Masamune from Porre and returned it to Denadoro Mountains, though that is only speculation (and that role could also be filled by Lucca who we know was still alive at that point).
What other facts do we know at this point? Really anything regarding Dalton or Crono/Marle right now is speculation based on personal opinions. If we focus on understanding the information on Porre given in Chrono Cross we might be able to find clues to some of our unknown questions.
On a personal note I would say the death of Crono and Marle offscreen would be even more anticlimactic then Dalton, as they were stars of the previous game and their deaths would definitely carry weight. I feel that with no proof that Chrono, Marle, OR Dalton are dead then we can safety assume all 3 survived the war.