"What kind of a person" vs. "what kind of person"

Thinking in depth about it, either could be considered correct (i.e. neither is positively incorrect), and both are common. It's true that an article should often be used before a noun. Hence the following is incorrect:

I accidentally jogged the arm of person walking by.

It should be:

I accidentally jogged the arm of a person walking by.

Or, if you have already been introduced to that person:

I accidentally jogged the arm of the person walking by.

However, since phrases like what kind are referring to a type, not an individual, the article a is not mandatory. Consider the phrase what kind of dog. Dog is a type here, not a specific dog.

The question form complicates this problem. Let's start by looking at this kind of X and these kinds of X. There seems to be some sort of concord between the number of the determiner (i.e., this/these) and the number of X. That is, you would expect this kind of cow or these kinds of cows but this kind of cows is rare and maybe even ungrammatical for some people, even though logically, cows would make sense since it is commonly used when referring to the general group (e.g., cows give milk). Consider also cows of this kind not *cow of this kind.

This concord suggests that the determiner for kind is bleeding through and filling the determiner slot for X too, which is why the noun in X can be singular without needing a determiner as other singular countable nouns typically do.

Going back to the original question, I think when people say what kind of cow, they take what to fill the obligatory determiner slot for cow and the kind of becomes somewhat transparent. At some level, they are asking what cow. This would be somewhat analogous to the subject-verb agreement in cases like what kind of cows were used (cf. what kind of cows was used).

Those who focus on kind, may be more prone to see the determiner slot for X as unfilled and fill it with a, just as they may force number agreement between kind and was in the example above.

I suppose it depends on the intent of the question.

If you are asking about the type or group of a person, then you may skip the article: "What kind of person dress in sarong to go to work?". Although in this case, I would use people instead of person.

If you are asking about the character of the (specific) person in question, then I think the article is needed, e.g. "What kind of a person who would do such heinous thing to a puppy?"

My opinion: If the article doesn't improve the clarity of a question ("what kind of man would. . .?" vs. "what kind of a man would . . .?), the article is unnecessary, which to me makes it more a colloquialism or idiom than a grammatical defect.