If you noticed improper behavior at work, ranging from petty theft of office supplies to deliberate, willful manipulation of a company's financial statement, and you decided to report it to HR, should you do so anonymously?
Remember that you can always claim an anonymous tip after the fact, but you cannot remove your name from it once given. IT's safer to do things anonymously, and be willing to put your name on it later if need be (as in, you report an exec for misconduct, HR wishes to press charges, but have no actual witnesses/evidence).
There are a few things that affect the decision you choose
Is the problem something you want to simply report and say you've done your duty, or is it something where you want to be sure action is taken?
If you're reporting just because you feel it is the "right" thing to do, then an anonymous tip is a good way to do so without getting involved in the whole process. How HR handles it from there is up to them.
However if you want to ensure action is taken, it is better to report in person. An anonymous tip can often be disregarded as someone just out to cause trouble, and does not have the same credibility as someone risking their reputation and possibly job to make the same claim in person.
How confident are you in your HR Manager's ability to keep a secret?
Every HR manager I have worked with has been really good about keeping secrets, even when their boss asks. Just be sure you specifically tell HR that you want to keep your name confidential, as they might not immediately assume that is the case.
If you have a valid reason for doubting their ability to keep a secret and you want to do more than just report the problem and be done with it, then I'd suggest reporting anonymously. It would be best if you give them an opportunity to ask you questions, such as calling in person from a payphone or using a non-identifying email account, instead of simply doing something like leaving a letter on their desk.
How serious is the crime?
If it is something likely to attract a federal (or state) investigation, it is possible the HR manager will be forced to give up your name, regardless of what they'd prefer to do. If this is a possible scenario, you may want to report anonymously just because you can't be forced to share information you do not have. But these cases are pretty rare.
With something like theft from the company, I'd use a payphone from outside the building to report anonymously to HR. Even though every large company I've worked for has had a stated policy of not harassing whistleblowers, I'd never want to test that policy by letting the reporting get traced back to me.
The fact that a company WANTS to do the right thing doesn't mean that everyone in the chain will actually DO the right thing. You have no idea what inter-office politics might get triggered by the report, so anonymous reporting lets you not be in the target zone if the whole thing blows up.
Most big companies have an anonymous 3rd party managed whistleblower phone number and/or email address. If that exists, I would use it.
If it doesn't, I would determine how important it is to me. Stealing office supplies is petty and not worth the trouble. Cooking the books is. I would report those directly, after updating my resume.
There are times where reporting anonymously is not possible or the best course of action.
If you are reporting problems with financial numbers or criminal activity then you should do so in a way that it is documented that you are bringing the problem forward. This is to protect yourself. There have been several cases where the person reported it ended up being blamed as the person responsible for the wrong numbers. If you can document that as soon as you found the problem you came forward it can help protect you from being blamed for someone else's crimes. In the investigation it is possible that it will come out that you had knowledge of the crime. Being the one to bring it forward can protect you, though I would engage a lawyer immediately to be sure.
Further if there are victims of the crime you have a responsibility to those victims to stand as a witness for them. This may or may not be a legal obligation but it is a moral responsibility.
The consequences of accusations like these are serious so they should not be made lightly. But when doing so it is more likely to be in your best interest to do so in a way that protects you when it goes sideways. When criminal activity is involved it nearly always does.
In some cases you have an obligation to report adverse information to a security officer or compliance officer. If the people involved are working on government contracts not reporting the situation can cause major problems. The longer the problem goes on the bigger the penalty, and you might get swept up into it if anybody knows you could have reported it.
Big companies in the US have ways of protecting whistle blowers, small companies may not fully understand the laws.
Government agencies have an Inspectors general. Some companies have an ombudsman. These organizations are insulated from the rest of the organization.
An anonymous phone call or note slipped under the door won't be enough to guarantee action, you will have to give contact information.