I have a second computer on which i want to access my emails. I also want to be able to access on my old computer. It was possible but no longer is.

I have a second computer (the one I am using at the moment) on which I want to access my emails. I also want to be able to access on my old computer. It was possible but no longer is. I do not need to access the messages stored on the other computer, I just want to be able to use my googlemail on the second computer. I have Thunderbird on both computers but when I put my password in the second one I get an error message and a message is sent to the other computer saying someone is trying to access my googlemail account. This used not to be the case. I am anxious to be able to use the secong=d computer because the other one is old and becoming unreliable.