How to use Japanese causative sentences to say the following?
He had me not wash the dishes.
I believe that "彼は私に皿を洗わせませんでした。" is not the answer because its translation becomes "He has not made me wash the dishes.".
Using the terms from snailboat's link:
Not [force doing]
He didn't have me wash the dishes (but I washed them because I was bored).
Similar to → He did not force me to wash the dishes.
Force [not doing]
He had me not wash the dishes (because I'm really clumsy).
Similar to → He forced me to not wash the dishes.
Verbs in the form 〜せなかった/〜せませんでした are ambiguous and can work for both of these two meanings. It will only mean one or the other depending on the context. You will need to rephrase it if you want to be absolutely clear. There are many options, but just to give a few:
Direct translation of "He had me not wash the dishes." is, as you wrote "彼は私に皿を洗わせませんでした。". That is correct.
But it's bit awkward for me, I real situation he stopped me to wash dishes because I wanted to wash. That was indicated, isn't it?
For me "彼は私に皿を洗わせてくれませんでした。" sounds more natural.