$\because$最简根式$\sqrt{a+3}$和$\sqrt[b+1]{3a-1}$是同类二次根式，
$\therefore \left\{\begin{array}{l}{a+3=3a-1}\\{b+1=2}\end{array}\right.$，
解得$\left\{\begin{array}{l}{a=2}\\{b=1}\end{array}\right.$，
$\therefore a+b=2+1=3$.
故答案为：$3$.

$\because$最简根式$\sqrt{a+3}$和$\sqrt[b+1]{3a-1}$是同类二次根式，
$\therefore \left\{\begin{array}{l}{a+3=3a-1}\\{b+1=2}\end{array}\right.$，
解得$\left\{\begin{array}{l}{a=2}\\{b=1}\end{array}\right.$，
$\therefore a+b=2+1=3$.
故答案为：$3$.