confirm computer? safe?

I use my computer for my banking, etc.(www.failbank.com)
I have to type in my password to log on to my bank site.(jimbo)
then I have to type in security questions because I don't have my computer "Versified" (1921, blue, 5-inches, but thick)
usually 2 to 3 questions, just so the site knows it's me and not you.

If I click the "YES, remember this computer", would that harm my computer safety? Make it more likely someone would get my password(jimbo)

I use ad-ware and ccleaner daily, so those might erase any point of me verifying my computer.