I'm of the belief that if a person gets a job done correctly and without causing any damage or offense or doing anything illegal, let them behave however they'd like on the job. That includes checking non-work-related emails. There is nothing wrong here as long as they do their job well.
If an employee has no access to outside email, then it could hurt how productive they are. If an employee is worrying over an e-mail that they may receive or may just have to receive some type of important e-mail, it could majorly hurt productivity from the stress related to this.
No, I do not think that it is alright to check emails that are not related to your work, because when your are at work it is to be productive, and to work your hardest to get the job done that your company needs to be completed by you at that time.
When you go to work for a company there is an understood agreement that you will provide a full days work in return for a full days pay. You would not want your employer to short your check, pay you late, or forget that you have vacation time coming. You would not like being cheated. But when we check personal emails while on the clock, that's exactly what we are doing - cheating the company.
I do not believe it is appropriate for an employee to check e-mails not associated with work while on the clock. I also do not believe it is appropriate to text other people or spend time on social media while on the clock. I believe when you are working, you should be doing tasks associated with your work.