After enabling Two-step verification for your LinkedIn account, you need to make sure your computer or mobile device you use to login to LinkedIn should be added as trusted device. its, very simple, here is how to do it.
Unlike to Twitter login Verification, Google and Dropbox does offer ‘trusted computers and devices’ option, so that you can mark or select your computer as trusted and from then onwards, you don’t need to enter verification code every time when you logon to your account. LinkedIn offers this option too.
Sign into LinkedIn, on verification code entering page make sure “Recognize this device in the future” is ticked. If you’re browsing from public or shared computer uncheck it.
Note: There are other situations, where you do get verification codes even though you login from your (Trusted PC) computer. This happens when cookies are not enabled for your browser from which you’re trying to sign into LinkedIn. Enable cookies and you won’t get those messages.