A general bound that's sometimes useful is given by Fernique's theorem. It's a general fact about Gaussian measures on Banach spaces, but in this case it gives the following: there are constants $C, \epsilon > 0$ (depending on the dimension $n$) such that $${\mathbf P} \left\{ \max_{t\in[0,2]} \|W(t)\|> x\right\}\leqslant C e^{-\epsilon x^2}.$$

You can find a proof in these lecture notes of mine.

A general bound that's sometimes useful is given by Fernique's theorem. It's a general fact about Gaussian measures on Banach spaces, but in this case it gives the following: there are constants $C, \epsilon > 0$ (depending on the dimension $n$) such that $${\mathbf P} \left\{ \max_{t\in[0,2]} \|W(t)\|> x\right\}\leqslant C e^{-\epsilon x^2}.$$

You can find a proof in these lecture notes of mine (see Section 4.2).