Is there a problem with playing Bananagrams on Shabbos because of Borer or any other reason?
Playing Bananagrams isn't a borer issue since you choose pieces for immediate usage. This is within the category of Derech Achila- normal consumption/usage. Even at the beginning of the game when the pieces are separated, this is normal immediate usage.
Kesiva (writing) isn't a problem since there is nothing to unify the letters (like a frame).
In some circles, adults (over 13) refrain from games and other mundane actions on shabbos.