My recommendation is to stay away from the Seine if you are looking for an inexpensive location. The outer arrondissements (9 through 20 excluding 16) all have lots of accommodations in every price range. The 9th and 10th arrondissements are the two arrondissements with the most hotels and yet do not feel very touristy because most of the people in the hotels are visitors from other parts of France or the nearest European countries (Gare du Nord and Gare de l'Est being the principal arrival point for trains from Great Britain, Benelux and Germany). In the 5th and 6th arrondissements, you would be more likely to be surrounded by aging Kansans with freeze dried purple hair.
A lot of people like the 18th arrondissement (the arrondissement with the greatest number of AirBnb rentals), particularly Montmartre, but quite a few others find climbing up and down the hills a bit arduous.
There is no "unsafe" location within the city limits, but some people seem to feel that "run down" or "sleazy" = "unsafe" so it is often a question of one's personal comfort level. Some people even find it unsettling that not all Parisians are of European stock, but that's a whole other can of worms. Since I live in one of the mixed areas, I just don't give any thought to the issue but quite a few people seem to obsess about it.