The expression is «avoir peur de» and in this case 'des' is not partitive, it is the contraction of 'de les examens' - that is 'of exams (in general)', not 'of some exams".  

In negations the definite articles le/la/les/l' do not change, and therefore de le/la/les/l' become du/de la/des/de l' . . Using le, la, l', les before nouns when generalising (definite articles)

In a case like this 'some' could be expressed by «avoir peur de certains examens» (afraid of some exams (but not all exams)), and «n'avoir pas peur de certains examens» (not afraid of some exams, (but afraid of others)) Avoir peur de = To be afraid/scared of (French Expressions with avoir)

Thank you Maarten. I guess I need a little more clarification.   It seems like “n’avoir pas peur de” is different from “n’avoir pas besoin de” as in “je n’ai pas besoin d’amis” meaning I don’t need friends (in general, not some friends)? Perhaps I just need to accept that “n’avoir pas peur de’ is different!

I am not sure I can fully explain it either! In brief if the noun is initially preceded by a definite article, that article carries through the construction into the negative. So - les examens - ends up at de les, and this becomes des. If however you start with a partitive article before a noun ie «des amis» the partitive article 'transforms' to de (or d') with the negative, hence «d'amis». So the 2nd sentence 'starts life' as «J'ai des amis» and builds in the negative etc from there, whereas the first starts as «J'ai les examens»

In the case of “Je n’ai pas besoin d’amis”, it's not important that it's negative, because it would be the same in the positive sense: “J'ai besoin d’amis”. The d' is the preposition "de" meaning "of", not the partitive. You don't get "des amis", because you can't say "J'ai besoin de des amis". The sound combinations of "de des" or "de de" etc. are never allowed because they sound bad. Sometimes this is called the "rule of cacophony", and it usually means that the combination is simplified to "de".

The difference between "avoir peur de" and "avoir besoin de" is that it affects the way you generalise about things. "Les examens" means something like "all the exams in the world", and you can be afraid of all of them. "Les amis" would mean "all the friends in the world", but you can't need all of them, you only need some.

Partially agree Alan, as per initial comment - however the combination of the conditions that definite articles remain after negation, and the 'de + definite article' contracts explains why it is 'des examens' ( in English similar changing 'the' or 'all' to 'of the' or 'of all') in the negative sentence. In the positive it is "les examens". It is also the negation rule (that partitive articles become de or d' - similar to changing meaning from 'some' to 'any' in English translation) that explains why «des amis» in the positive becomes «d'amis» in the negative rather than remaining «des amis»,  'J'ai des amis' to 'Je n'ai pas d'amis'. The question is why 'des examens' (not d'examens) when it is 'd'amis' (not des amis) in the negative constructs.  The cacophony rule only explains why it is not 'de des' - it doesn't explain why it is de/d' instead of des.

But the positive sentences that we are talking about don't contain partitive articles, so there's nothing that needs to be changed when forming the negative sentences. It's a different story with "Je n'ai pas d'amis", of course. In that case it is because of the negation.

"J'ai peur des examens." (de + les -> des)

"J'ai besoin d'amis." (de + des -> de -> d')

It's true that the cacophony rule doesn't always result in "de" - I think there are some  exceptions where you get "des" instead - but it's generally the prepositional "de" that is retained, and the partitive article disappears completely.

Thanks Alan. I am happy to accept I may be going down a rabbit-hole trying to explain the difference in the sentences in response to Judy's question. Looking at a number of other forums, I wouldn't be the first! Accepting that«des» can be partitive or indefinite, or a contraction, in terms of the discussion with negation it doesn't change the outcome. Going back to Judy's original question - do you think «je n'ai pas peur des examens» is correct? I think it is correct or at least regularly used, and have not found anything that shows it to be incorrect, but if it is incorrect then the discussion is moot. I am also fairly certain that 'n'avoir pas d'amis' is correct. If both are correct, do you know why 'n'avoir pas peur des . .' is acceptable but not 'n'avoir pas besoin des . .'.?    since both are expressions of the form (ne) avoir (pas) ..... de ?

I think that "je n'ai pas peur des examens" is correct, because it is formed, quite simply, by adding "ne ... pas" to the positive sentence that I gave. 

"N'avoir pas d'amis" is also correct - but this is the only case we've mentioned where the rule of changing partitive/indefinite articles in negative sentences to "de" is relevant. If "des" is the contraction of "de" + "les", it doesn't change in a negative sentence.

"N'avoir pas besoin des" is also possible, but only if "des" means "de + les". With "avoir besoin de", unlike "avoir peur de", this can only happen when talking about something specific, not a generality, so you can't say "je n'ai pas besoin des amis", because that would mean "I don't need the friends".