[彼は元気ですね]は正しいです.

- 元気がある　 　　He has a nature of ”元気がある" 　　Though sometimes he can be in the status of "元気がない", even then, he is still a "元気がある" person as a nature. 「彼は元気があるね」 implies that the speaker can see his nature deep inside, which sometimes can lead to sounding like "上から目線", a kind of arrogance. Having read「彼は元気があるね」sentence only , I have a impression that the speaker is a boss in a company and "彼" is his man. - 元気です 　　He is/has been in the status of "元気がある", regardless of his nature.

All of them are correct as far as my knowledge goes 彼は元気あるね is a contraction and means the same as 彼は元気があるね and means that "he is energetic/high spirited" 元気ですね means "he's doing good" and means that he is healthy and energetic. My source: http://pinasan.com/english/today_study/12073