This exact thing happened to me.
My shift account was made years ago, and I have not changed the password. So the password I used was 6 lower case letters, followed by four numbers. I was able to login fine in BL2 and all other games, and even the Shift website itself, with my ‘old’ password. Sometime recently, the Shift Password requirements changed, requiring Upper case letter, Lower case letter, a number, and a non alpha-numeric character. Since my old password did not contain these characters, BL3 just assumed my password was incorrect without even checking it. As soon as I changed to a newer password (that meets present requirements) the ‘login’ button was no longer grayed-out in BL3, and I was able to sign in without problems.
Long story short : People must use a new password with SHiFT that includes all those listed characters.