Avoid anywhere in or around Montmartre, in particular Pigalle and Barbes. Most neighbourhoods south of the Seine before the highway (the Peripherique) are safer, but also more expensive. Porte d'Orleans is supposed to be pretty good and the Quartier Latin is cute and vibrant (and close to Notre Dame...but also probably more expensive). There may be some chain hotels (so maybe less expensive?) just on the other side of the highway south, but don't go too far past because then you're in the suburbs and transit gets to be a pain.
(this is advice I got from a man from France and a woman who lives in Paris)
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