"I don't take taxis."
Both に, は, and には work here.
に would be the one used normally in the positive sentence.
は can be used in the negative to stress that taxies are not what you're taking (probably usually replying to a question)
には can be used to stress the "in a taxi"/"on a taxi" part (also probably replying to a question, but I think that not necessarily)
I may be wrong, though. If I am, I hope somebody will correct me
Well; のる is to "get on" a vehicle (car, train, plane...) or to travel by it ("ride" it), and つかう is "to use", in a more generic way. Then, the difference would be the same than between "I use a taxi" and "I travel by a taxi". In this sentence both are valid, but the verbs themselves have different meanings when alone.
使わない is also accepted for this question, and I was wondering if there are any implications to choosing 乗る vs 使う when talking about using vehicles for transportation beyond the English translation of ride/board/take vs use, specifically in the negative like this sentence. Are there contexts in Japanese where you'd definitely want to use one over the other, or are they as interchangeable as they seem in English?
I had to remind myself that 乗る is a U verb rather than i verb and found this list of verb conjugation for 乗る(noru) handy http://www.japaneseverbconjugator.com/VerbDetails.asp?txtVerb=%E4%B9%97%E3%82%8B
A Taxi is not something that you would not take as a whole, rather it is the act of the taxi ride that one does not...
に links the object into an action, with, I do not ride with a taxi は applies the action to the object , is, A taxi is not for my ride には makes the whole action of not taking a "taxi-ride"