A

$\because$单项式$3x^{3}y^{2}$与$-4x^{a}y^{b-1}$的和仍是单项式，
$\therefore 3x^{3}y^{2}$与$-4x^{a}y^{b-1}$是同类项.
$\therefore a=3$，$b-1=2$，解得$b=3$，
$\therefore a^{b}=3^{3}=27$.
故选：$A$.