The only one under contract was Grace Lee Whitney, as a regular (a contract which was promptly nullified when she had her...problems). The rest were day players. Walter Koenig has said in interviews that he was never under contract.
Koenig was hired as a day player, but according to his autobiography, he was upgraded to a contract player at the beginning of the third season for 7 out of every 13 episodes (Warped Factors
, pg. 194).
I've seen the documentation from the first season in the Roddenberry archives at UCLA which shows that the others were contracted as well (except for Nichols), at least for the first 13 episodes of the series. Here's, in part, what it says ("First Production Year 1966-1967," dated May 31, 1966):
William Shatner -- Guarantee: 13 out of 13 including Pilot
Leonard Nimoy -- Guarantee: 13 out of 13 including Pilot
DeForest Kelley -- Guarantee: 7 out of 13 including Pilot
Grace Lee Whitney -- Guarantee: 7 out of 13 including Pilot
George Takei -- Guarantee: 7 out of 13 including Pilot
James Doohan -- Guarantee: 5 out of 13
That information and more is reprinted in David Alexander's biography of Roddenberry. The book is mostly awful, but it provides useful tidbits like this.