I should say either "Alcohol is harmful to health" or "Alcohol is bad for the health."

Why not "the health"?

Different kinds of people use different words. The same people use different words in different situations. The words and patterns that are special to situations or social or educational levels are called registers. Formal, written English is a register. Slang is another register.

"Is harmful to" belongs to one register (it's fairly formal): "is bad for" belongs to a different (less formal) register. "Health", with no "the", is very general: anybody's health, everybody's health." Adding the definite article to get "the health" makes it particular, intimate: now it means your health, my health, our (just the two of us) health.

So. When we say, "is harmful to," we are speaking in a formal, impersonal register where we naturally speak of health in general: no "the." When we say, "the health," meaning yours and mine, we are speaking in an informal, intimate register where "is harmful to" is inappropriate.

You will never be able to consciously apply this kind of analysis to decide when to use "the" and when to leave it out. No English speaker does that. Instead we unconsciously absorb patterns, whole phrases that are stored in some mysterious way in the speech centers of our brains and guide the word choices we make.

This is the reason it takes so long to learn a language--including one's native language. The brain must construct a huge number of subtle patterns and filters.

It's also the reason that most non-native speakers stick out like a sore thumb: the things they say don't fit, they don't get through all those filters we have constructed.

@halfb1t
Interesting article. Thank you!