These are the playable character with whom Balthier travels throughout the game. With some he gets more indepth interaction than with others, but this is my take on their relationships. Fran has her own section as his partner.

Vaan is our perspective in the game. Alot like him, we haven't seen much of the world of Ivalice before the party sets out on their journey.
Despite being an orphan, with the memory of his brothers death from two years ago still fresh in his mind, Vaan manages to keep his chin up. I'd say he has a pretty "No problems, just leave it to me"-attitude which very well complements Balthiers more stoic approach to matters. In the same way, Balthier is cool while Vaan is hot-tempered. The two can learn alot from eachother.
In the beginning Balthier only uses Vaan as an extra pair of hands, needed in order to survive through the sewers of Rabanastre. Yet when Vaan is beaten by the Bangaa at Nalbina Fortress Underground Prison, Balthier comes to his rescue and asks "You alright?" in a silent voice which could easily be thought to contain concern.
Vaan's admiration really kicks in when he sees the beautiful Strahl, Balthier's airship. As a sky pirate Balthier is living Vaan's dream and even though he promptly denies it, he is the boy's mentor in the trade.
I believe the respect goes both ways. Whenever Vaan has accomplished something or gets a good idea, Balthier makes sure to praise him.

Upon their first meeting princess Ashelia B'nargin Dalmasca was introduced as Amalia, a member of the resistance. Little did Balthier know that she would turn out to be the sole heir to the Dalmascan throne, basically; More trouble than he had bargained for.
She is a princess different from the once we are used to seeing, a princess hardened by the times of war. Yet her strong and somewhat harsh and angry surface reveals alot of insecurity. Ashe struggles with her desire to obtain the nethicite's power, in order to restore her kingdom. This gives Balthier a reason to mistrust her but in the end he let's it go and decides to believe in her strenght.

One is 22 years of age and the other 36, yet they speak to eachother like equals. Although different in professions, I believe they share a mutual respect for one another. At first encounter Balthier totally ignores the poor, caged knight but soon after he becomes the first person to believe that Basch didn't kill his own king.
They are both wellinformed and have quick reactions in hostile situations, which makes them a good team. One example is when they take down an imperial guard together after being taken aboard the Leviathan. Balthier snatches his weapon giving Basch an opening to hit the guard in the back.

They had most interaction in the beginning of the game. Balthier shows some of his courtly, gentleman sides when speaking to Penelo. As Balthier, Fran and Vaan were about to be thrown into the Nalbina Fortress Dungeon Prison, he gave the scared and sad Penelo his hankerchief, assuring her that he would bring back Vaan soon enough.
When eventually she gives the hankerchief back, in the english version he gives her a little bow and says he will wear it close to his heart.
He is very nice and charming towards her, but I don't know if he sees her as an important partymember or just as "Vaan's girl". But as himself and Fran, Vaan and Penelo are often viewed as a pair. Balthier once referred to them as entertainment. They are vital to the group, not only because they develope into good fighters, but because they releave the journey of it's tension, making it alot more fun!