Doctors certainly feel dejected if they are being drawn attention for their hygiene. Being a healthcare provider, I know this. That's because advising something about their profession (how to examine a baby does include washing hands with sanitizer and that's what is taught to all child specialists) is nothing less than advising fish how to swim. It's overreacting but it's the truth.
Now, how you do that?
There is one good way which won't put anyone (a doc and you) in humiliation. There should be a feedback/suggestion box. Write it down (anonymously) and just drop your suggestion. If the management is active enough (I hope it is!), it'll be conveyed in a very straightforward way. You asking something to do and the management advisesadvising to do it makes a lot of difference. Alternatively, you can also put an anonymous mail to the organization with polite words (no offense at all) starting with In the interest of patients and organizations, we'd like to draw your attention to a trivial matter which could turn out to be a major health risk..." and so on...