Paris stay

Basically, all of Paris is safe. With a budget of at least 200€ per night, you should have many good options. Popular areas include the area between les Invalides and le Jardin du Luxembourg (mainly the 6th arrondissement), the 3rd, 4th (le Marais), and 5th (most of the Latin Quarter). Hotels in the 1st, 2nd, 8th, and 9th also offer a great deal of interest as well.

You might do an internet search of hotels along rue Jacob in the 6th. The area is very interesting, and the hotels offer what many visitors want - a central location with a good amount of old world charm.

My favorite area is Ile St Louis (in the 4th arrondissement), the little island next to Notre Dame. If you look on a map it’s the most central area of Paris and eminently safe. I recommend Hotel de Lutèce or Hotel des Deux Iles.

For a first visit you want to be in the center which is an easy walk to most of the sights you will want to see and accesses good transport for the rest. The Eiffel Tower is on the edge of the city and not convenient. I'd go with the 4th, 5th or 6th near the river -- the Marais, Latin Quarter, St. Germain. I have never stayed on Ile St. Louis but it is of course very central although there is no metro on the island. With the current Bataclan trial and the destruction of Notre Dame -- it is not quite as nice a setting as in the past.

AirBnBs aren't illegal but Paris requires owners of short-term rental properties to register and pay a tax. Look for a registration number on any listing, and if there isn't one don't go further. That said, with only three nights for a couple I'd suggest a hotel. Simpler to check in and out, a front desk to help you if necessary, a short stay. Your budget gives you room for good choices.
Good suggestions above from people who know Paris well. Ask for a room facing away from the street if possible, for quieter nights.

One issue to keep in mind - You don't mention which month you will be in Paris . Be aware that September is a very busy time of year , and hotels book up well in advance . I will be there in September of 2022 , and many hotels are already booked and prices tend to be high then .

We'll be there in June

The trial is not really affecting Île -St-Louis. The trial is on Île de la Cité and has not really been having a big impact on the other island. Really any hotel whether in the 5th or on the island that has a view of Notre Dame has been impacted by the fire subsequent rebuild so I would not let that stop me from staying on Île-St-Louis if I were so inclined. I happen to think that Le Marais and Saint Germain are the best areas to stay for a 3-day first visit with the Latin Quarter and parts of the 1st arrondissement being other great options. June is right up there for room rates. What currency is "250 a nightish?" If it is USD, that is good budget and you should have some good choices among hotels even in June. Here are two to get you started: https://hotelparisodeonsaintgermain.com/en/ and https://www.hotelcaron.com/en/

AirBnBs aren't illegal but Paris requires owners of short-term rental properties to register and pay a tax.

The rental situation is a lot more complex than simply registering and paying a tax. Very broadly, legal short term rentals are limited to either: the apartment of an owner/occupant or commercial apartments of which there are very few, around 200 in all of Paris. Second homes or vacation apartments are illegal to rent short term, even though many of these owners have applied for registration numbers.

Almost anyone staying in Paris fewer than 30 days should primarily consider either apartment hotels or regular hotels for their stay.

I recommend the Hotel Residence des Artes. https://www.hotelresidencedesartsparis.com/?gclid=Cj0KCQiA7oyNBhDiARIsADtGRZY38JwD2C5hh14GIS7GLy6KT1OYB5S5nPpLg9O5rxwS1q6xSjDOYWwaAldMEALw_wcB

Ideal location in St. Germain, the oldest part of Paris. A block from the Seine, directly across from Notre Dame. Charming streets, tons of shops and restaurants. Several metros stops and RER within a block. The double rooms (194€/nt) are a good size for Paris and include a mini fridge. The Jr Suites are double the size and feel palatial for Paris. But I see they are 288€/nt in June.

I would not stay near the Eiffel Tower. It’s out of the way for the majority of other sites you will want to visit.

We enjoyed staying in Le Marais here: https://www.carondebeaumarchais.com/index-en.html We loved the vibe of the area, charming hotel, short walk to Notre Dame, the
Seine, a block to metro, great breakfast, and within your budget- €130-250/night. We would stay there again. We felt totally safe everywhere in Paris. We took a walking tour of Le Marais and Montmartre with Sightseekers Delight. The guides were knowledgeable and personable.
Edited to add: https://sightseekersdelight.com/about/

"I love it when the competition starts about the different areas. This's the only reason I made a suggestion because I know that the rest of you will fight it out as though you might win a prize."

Does this mean that the République suggestion is not a real one and was made only to generate this supposed competition.

I imagine you do love it kerouac.

Having just been there, Ile St Louis is not affected by the trial or work on Notre Dame, and is lovelier than ever with so few Americans around.

I agree with JHK, the Marais (in the 4th) is fantastic.