My keyboard stops working once PuTTY displays the password prompt.

PuTTY

A.7.12 My keyboard stops working once PuTTY displays the password prompt.

No, it doesn't. PuTTY just doesn't display the password you type, so that someone looking at your screen can't see what it is.

Unlike the Windows login prompts, PuTTY doesn't display the password as a row of asterisks either. This is so that someone looking at your screen can't even tell how long your password is, which might be valuable information.