According to Hesiod, by the time the Olympians came to power, Golden man had vanished. Hugh G. Evelyn-White's translation of Works and Days (from ellopos.net) describes them as
(ll. 109-120) First of all the deathless gods who dwell on Olympus made a golden race of mortal men who lived in the time of Cronos when he was reigning in heaven. And they lived like gods without sorrow of heart, remote and free from toil and grief: miserable age rested not on them; but with legs and arms never failing they made merry with feasting beyond the reach of all evils. When they died, it was as though they were overcome with sleep, and they had all good things; for the fruitful earth unforced bare them fruit abundantly and without stint. They dwelt in ease and peace upon their lands with many good things, rich in flocks and loved by the blessed gods.
They were loved by the gods, but nothing is mentioned about man worshipping or serving the gods.
It seems it was Zeus who first demanded that of the Silver man (same source):
(ll. 121-139) [...] But when they were full grown and were come to the full measure of their prime, they lived only a little time in sorrow because of their foolishness, for they could not keep from sinning and from wronging one another, nor would they serve the immortals, nor sacrifice on the holy altars of the blessed ones as it is right for men to do wherever they dwell. Then Zeus the son of Cronos was angry and put them away, because they would not give honour to the blessed gods who live on Olympus.
And then on it goes to the Bronze and finally the Iron age.
At least according to Hesiod it seems the Titans didn't need worhip, at least from the Golden man.
As for real people worshipping Titans, I can't think of a culture that worshipped Titans exclusively, but Cronos had a temple at the foot of the Acropolis in Athens, so at least one Titan was worshipped by real people, as cosmovision states:
Cronos ne figure guère que dans les mythes théogoniques; pourtant il avait un temple à Athènes au pied de l'Acropole.
Cronos hardly played a part in mythology except in theogenic myths; and yet, he had a temple in Athens at the foot of the Acropolis.
And of course there is the temple of Saturn in Rome (Wikipedia) and the Roman Saturnalia (Wikipedia), as well as the day after Friday. Saturn is usually seen as the Roman association of Cronos, so at least that one Titan did get worshipped quite a lot by real people, although, admittedly, it seems never to exclude the Olympians.