Why did the Valar not take action when Sauron's power began to rise? Why was it left to the residents of Middle-Earth?

Seriously??  The Valar were constantly meddling with Sauron in Middle-earth ... the poor sap never had a chance.

Evidence of the Valar's nearly non-stop interference with Sauron's plans for world domination can be found scattered throughout The Hobbit and The Lord of the Rings.  Here is a mere sampling of the more obvious instances:

The Istari.  Every answer here has already mentioned this, and I will too.  But I'll go even further and identify the other significant person who was also sent from Valinor to Middle-earth to oppose Sauron:

Glorfindel, my absolute favorite character in all of Tolkien's canonical works.  Who, not coincidentally, played a pivotal role in ending the menace of the Witch-king of Angmar's realm in Eriador without which the Dúnedain would have been extinguished.  And who, not coincidentally, rescued the bearer of Sauron's Ruling Ring from his servants the Nazgûl.  

The Eagles.  Everyone's (least) favorite deus-ex-machina plot device, Gwaihir the Windlord, his brother Landroval and the rest of the Eagles of the Misty Mountains were direct servants of the Valar.  They were descended in an unbroken line from the mighty Thorondor, messenger of the Vala Manwë himself ... and possibly a Maiar spirit in eagle form.  Not only did the eagles save Bilbo, Thorin & Company's bacon more than once on their quest, Gwaihir personally assisted Gandalf out of numerous jams (e.g., captivity at Orthanc, transport to Galadriel at Lothlórien, saving Frodo & Sam at the Cracks of Doom).

Bilbo Baggins' unbelievable luck.  The mathematical odds against a stay-at-home hobbit from the Shire wandering into the Misty Mountains to find Sauron's Ruling Ring are astronomical.  He was obviously a pawn of greater forces who merely acted out a divinely-appointed role designed for him by the Valar.  Gandalf the Grey admits as much to Frodo deep within Khazad-dûm:

“... [the Ruling Ring] abandoned Gollum. Only to be picked up by the most unlikely person imaginable: Bilbo from the Shire!
‘Behind that there was something else at work, beyond any design of the Ring-maker. I can put it no plainer than by saying that Bilbo was meant to find the Ring, and not by its maker. In which case you also were meant to have it. And that may be an encouraging thought.”

Boromir's and Faramir's visions.  As if that wasn't patently obvious enough, the Valar basically make direct mind-to-mind contact not once but several times with Denethor II's sons Faramir II and Boromir II.  The latter relates at the Council of Elrond:  

“... a dream came to my brother in a troubled sleep; and afterwards a like dream came oft to him again, and once to me.  In that dream I thought the eastern sky grew dark and there was a growing thunder, but in the West a pale light lingered, and out of it I heard a voice, remote but clear, crying:
Seek for the Sword that was broken:
In Imladris it dwells;
There shall be counsels taken
Stronger than Morgul-spells.
There shall be shown a token
That Doom is near at hand,
For Isildur’s Bane shall waken,
And the Halfling forth shall stand.”

Frodo's decision to be the Ringbearer.  At the Council of Elrond, Frodo wanted nothing more than to wash his hands of the whole Ring business and stay safely in Imladris.  But the Valar had other plans for their pawn:

"An overwhelming longing to rest and remain at peace by Bilbo’s side in Rivendell filled all his heart. At last with an effort he spoke, and wondered to hear his own words, as if some other will was using his small voice.

‘I will take the Ring,’ he said, ‘though I do not know the way.”

Gandalf the Grey's resurrection.  While we can't pin this directly on the Valar, since even they probably lack the firepower to directly resurrect a slain spirit, at the very least they put in a good word for Gandalf with Eru Ilúvatar, the Man Upstairs.

Rejection of Sauron's spirit.  Even after manipulating events to guarantee Sauron's downfall and destruction the Valar added insult to injury by showing their contemptuous disfavor by personally banishing his spirit from returning to Valinor.

"The realm of Sauron is ended!' said Gandalf.  The Ring-bearer has fulfilled his Quest.'  And as the Captains gazed south to the Land of Mordor, it seemed to them that, black against the pall of cloud, there rose a huge shape of shadow, impenetrable, lightning-crowned, filling all the sky. Enormous it reared above the world, and stretched out towards them a vast threatening hand, terrible but impotent: for even as it leaned over them, a great wind took it, and it was all blown away, and passed; and then a hush fell.”  [Note:  according to Tolkien's map of Mordor, this wind would've come from a northwesterly direction ... almost directly from Manwë's palace upon Taniquetil.  Manwë, natch, was the master of winds.]

Honestly, the Valar's repeated interventions had Sauron knee-capped from the get-go.  Gandalf made this plain to Bilbo Baggins long before the War of the Ring even started.

“You don’t really suppose, do you, that all your adventures and escapes were managed by mere luck, just for your sole benefit? You are a very fine person, Mr. Baggins, and I am very fond of you; but you are only quite a little fellow in a wide world after all!"

You know who's not little fellows?  The Valar.  They tied up Sauron in their strings of destiny and made him dance like a marionette, the poor bastard.