Some parts of the Apollo project were started under the Eisenhower Administration. This can be proven by looking at the Apollo Project Timeline on NASA's website. For example:

1959:
January 19: Contract signed by NASA with Rocketdyne Division for design and development of the F-1 engine

It was 5 of those F-1 engines that powered the Saturn V first stage.

December 31: NASA approval of the Saturn C-1 configuration and the long-range Saturn development program.

1960:
April 26: NASA selection of the Douglas Aircraft Company to build the second stage (S-IV) of the Saturn C-1.

May 31: Selection of Rocketdyne Division by NASA to develop the 200,000-pound-thrust J-2 rocket engine

July 28-29: The announcement of the Apollo program to representatives of American industry.

September 1: The Apollo Project Office formed under the Space Task Group (STG) Flight Systems Division.

This early start on the Saturn V and Apollo project were essential to our being able to land on the moon before 1970. That said, it was Kennedy who lit the fire and got things moving.

By today's standards, KFK was a fairly conservative Democrat. He was instrumental in ramping up activity in Vietnam, in forming the Green Berets, and in pushing massive tax cuts. From his inauguration speech.

"Let every nation know, whether it wishes us well or ill, that we shall pay any price, bear any burden, meet any hardship, support any friend, oppose any foe to assure the survival and the success of liberty.

This much we pledge--and more."

and

"We dare not tempt them with weakness. For only when our arms are sufficient beyond doubt can we be certain beyond doubt that they will never be employed."

and

"In the long history of the world, only a few generations have been granted the role of defending freedom in its hour of maximum danger. I do not shrink from this responsibility--I welcome it. I do not believe that any of us would exchange places with any other people or any other generation. The energy, the faith, the devotion which we bring to this endeavor will light our country and all who serve it--and the glow from that fire can truly light the world."

This doesn't sound like too many of today's Democrats. While Kennedy spoke so well of his desire for peace, he knew that we had to have a strong military and the will to use it.