I've worked both ways and I'm unsure which way is best. There are clearly benefits to each way.
You already understand the benefits of 'blind' marking - you are more likely to be unbiased. However, by knowing the student behind the writing there are some benefits (along with the danger of bias).
One benefit is that you can authenticate more easily. That is, if you keep reading papers from one student and you see a sudden jump in quality (up or down) it can alert you to the possibility the work is not theirs (hired writers, etc.). Another benefit is that you can watch the development of each student over time. This is a little more difficult if you do not know while you are marking. Of course, you can go back over the papers after you submit the grades but that requires double reading (and not many want to do that for each student). There is the added benefit of seeing something which jumps out, like a special quote or comment you made in class. You can see who is getting it and who is not while you are marking. This last point is less of an issue because for stand-out cases you are more likely to read twice and it would be less work since it's a one-off event.
For exams, there is less of a benefit to the authenticity point above (since they were in the exam room) but it does still allow you to keep an eye on students. There is the obvious administrative benefit of not having to anonymize the exams.
As for preventing bias when papers are not anonymous, I consider each mark quite carefully and ask myself if I am being biased. When I look over all the marks and I see that I passed students I cannot stand and that I failed students I really like, then I feel fairly confident that I am keeping my bias in check.
