Well, we have an obvious upper bound of TA 2988 (from Appendices A & B):
2988 Finduilas dies young.
‘After [Finduilas'] death Denethor became more grim and silent than before, and would sit long alone in his tower deep in thought, foreseeing that the assault of Mordor would come in his time. It was afterwards believed that needing knowledge, but being proud, and trusting in his own strength of will, he dared to look in the palantír of the White Tower.
We have a lower bound, also from Appendix A, of "before his sons grew to manhood":
‘In this way Denethor gained his great knowledge of things that passed in his realm, and far beyond his borders, at which men marvelled; but he bought the knowledge dearly, being aged before his time by his contest with the will of Sauron. Thus pride increased in Denethor together with despair, until he saw in all the deeds of that time only a single combat between the Lord of the White Tower and the Lord of the Barad-dûr, and mistrusted all others who resisted Sauron, unless they served himself alone.
‘So time drew on to the War of the Ring, and the sons of Denethor grew to manhood.
His sons were born TA 2978 and 2983, so it's likely that his first viewing through the palantir took place between TA 2988 and 2998. The reason I give them 20 years for manhood is because that's when Aragorn was adjudged to have entered his manhood as a slower aging Dunedain by Elrond:
'when Estel was only twenty years of age, ... Elrond looked at him and was pleased, for he saw that he was fair and noble and was early come to manhood.
So that ten year gap is the most likely case for when he started using it, triggered by the death of his wife. Interestingly, that actually puts him earlier than Saruman and his entrapment:
c. 3000 Saruman dares to use the palantír of Orthanc, but becomes ensnared by Sauron, who has the