Do banks really have people on the boards of corporations, and why? I read somewhere that in the United States, there are often people from banks on the boards of executives of corporations. Note, I don't mean ex-bankers who now work for that corporation, I mean people who are actually employed by the bank, and who are on the board in their capacity as a representative from the bank. 

This surprised me very much, since the board of directors in principle represents the equity-holders, not the debt-holders of corporations. 

How often does this really happen?

Why do banks do this, and why do corporations allow this?
What you describe is neither unusual nor pernicious, and occurs more commonly at smaller companies when the company has borrowed significantly from a single bank. Larger companies have direct access to capital markets (in addition to stock, they can issue bonds and commercial paper, which are both forms of debt), and are thus less reliant on their relationship with a single bank.

When companies are reliant on a relationship with a bank for access to debt, the bank will often seek a position on the board as a way of monitoring the activity of the company. From the bank's perspective, this is totally reasonable, but to understand why, you first have to understand a tiny bit of industrial organization theory.

In a nutshell, firms have what we refer to as "capital structures," which are a way of allocating risk and return. 

At the top of capital structures is senior debt, which is first in line to be paid, but which receives the lowest (though a fixed and predictable) rate of return. If you hold senior debt, you want the company that issued the debt to take little risk, as you'll be paid the same amount so long as the company remains solvent. If the firm goes into bankruptcy, you'll still be paid first, but your money may be tied up for a time in bankruptcy proceedings, and you face having to reinvest the cash you get back, potentially at a lower rate of return. As a result, you want to avoid unnecessary risk.

At the bottom of capital structures is equity, which is high-risk and high-return. The equity portion gets paid the upside if a firm does well, and it bears the first loss if a firm performs poorly. If a company goes bankrupt, the equity is worthless. Equity investors have a lot of incentive to take on risk, because for each dollar invested, the downside is a maximum of a dollar (this is known as "limited liability"), but the upside is limitless.

In between these two parts of the capital structure, there are all kinds of hybrids of debt and equity, but we can set them aside for simplicity. Most important is that there is a fundamental conflict between debt holders, who want limited risk and who only have control of a firm once it is bankrupt, and equity holders, who favor risk and control the firm in the normal state of the world.

Once you understand this conflict, it's easy to see why a bank would ask for a seat on the board of a company when it holds that company's debt: by having a member on the board, it can ensure ongoing, regular access to the information it needs to be comfortable that the firm isn't taking on too much risk. If it sees that the firm is moving in a direction that it isn't comfortable with, it'll decline to roll over its debt, getting its money back and forcing the firm to find financing elsewhere.

So: is this unusual? No. Directors who don't directly "represent" equity holders aren't at all uncommon. In fact, the majority of directors at large firms in the US are "independent" directors who have no direct relationship to the firm, other than serving on the board. Many non-independent directors are "executive" directors, who are company management— a group that, like creditors, may have incentives that differ from those of shareholders. Other groups, like labor unions, are also sometimes represented on boards. So if in addition to equity holders, outsiders, management, and unions are all represented on boards at times, what is the specific objection to creditors being represented?

It's additionally worth noting that the only case where a creditor's vote matters is solely when the other directors are evenly split on an issue. If a particular matter clearly benefits creditors at the expense of equity holders, a significant fraction of equity holders would have to vote against their own interests for this to be the case.

Finally, to your question as to why regulators and corporations allow it. Simply put, regulators have no reason (from a market conduct, safety and soundness, consumer protection, or financial stability perspective— the general scope of regulators' mandates) to object to this sort of arrangement. For a regulator to intervene in this way would interfere with the company's freedom to contract however it chooses. Creditor board members are not a thing that banks do to corporations, who passively allow it; rather, the boards of directors have to vote to add the board member representing the creditor. So if one views positively the judgment of a board composed of equity holders, why should one question the judgment of that board when they vote to add a creditor as a member?

So in summary: banks as board members is a way of monitoring/limiting risk-taking by firms, firms "allow" this because they want to borrow money from the banks, whether it is socially optimal depends on whether firms should be taking more or less risk, and it's more about correcting information asymmetries than about asserting day-to-day control— boards of directors don't tend to have lots of measures passing or failing by a single vote.