Ballad - Definition, Meaning & Synonyms | Vocabulary.com
A Very Strange Question

Ballads can be played on any saxophone (well maybe not sapronino lol).  The question you need to answer is what horn do you prefer to hear ballads play on.  
When I think of sound quality, it is more dependent on a players oral cavity and embouchure, mouthpiece choice, and reed.  The actual horn generally has less influence on sound than those aforementioned factors.  
The bad news is that you can't just go buy a horn and get the sound you want.  You have to make the sound you want through practice. 

Pete, that is a rather open ended question, because all saxes can play ballads. It is mostly up to the artist whether they want to use soprano, alto, tenor, baritone, or other sax.

What one do you feel comfortable with? And, the mouthpiece plays a big part in it too, having the one that gives the timbre you want.

I've used all of the above to play ballads. If you are already a sax player, why not just go with what you have now? After practicing ballads for a time, then you will have a better idea of what type of sax you'd want to use in ballads.

Warm and expressive sound: Known for its sweet and lyrical tone, the alto sax is a popular choice for ballads. It allows for subtle phrasing and nuanced dynamics, ideal for conveying emotion in slower tempos.

Classic ballad sound: Many iconic ballad recordings feature the alto sax, making it a familiar and evocative choice for listeners.

Responsive and manageable: The smaller size and lighter keywork of the alto make it comfortable and responsive for delicate playing, suitable for intricate melodies and smooth legato lines.