Good hotels in Nice that are less than 200 euros a night?

When are you going? How many nights? That may make a difference in hotel prices and recommendations as rates may be higher and some hotels have minimum number of nights in high season. There are lots of hotels in Nice <€200/night and many are within 'walking distance' of the sea, depending on what you mean by walking distance. I stayed on the pedestrian street close to the main square and old town and was a few blocks walk from the promenade along the beach. I stayed at Hotel Solara in July and it was <€100/night. That was a single room but I'm pretty sure the doubles are <€200/night.

Time frame and number of people is critical; that's an adequate budget to get what you need. For example, the Windsor Hotel or the Hotel Suisse. There are a number of hotels near the Old Town or in the Carre d'Or (roughly bounded by the Promenade des Anglais, Blvd Gambetta, Blvd Victor Hugo and Jean Medecin) plus the few blocks around Place Massena that fit the bill. If you are there for more than a few days, consider a managed apartment as well.

June, 3 nights, just two people. We just plan to visit the coastal towns.

We're on an RS tour that ends in Nice at end of May. We're staying a few extra days and decided on the Hotel Albert 1. https://www.hotel-albert-1er.com/en/
I can't actually vouch for it yet, but I do tend to research things to death before making a decision. My priorities were near the water and above average but not luxurious. For this occasion we chose a room with an ocean view for above 200 Euros/night, but they do have plenty under your budget as well. We don't tend to stay at high end hotels as we are usually out exploring for 10-12 hours a day and don't feel the need to spend for amenities we likely won't use.

We stayed at Splendid Hotel & Spa a few years ago -- 7-minute walk to the sea, rooftop pool, okay rooms, and generic decor.

I was looking a couple of weeks ago and found a lot under $200 on booking.com.
It’s pretty late to be looking for June so reasonable prices may be gone, but i’d look there.

I will be staying at the Hotel Regence for $641 for 5 nights. The reviews are decent for a budget hotel except for the breakfast, but I was planning on walking to breakfast, anyway.

We stayed at Hotel Vendome in Sept for just under $200 a night. Check booking.com. We were very pleased with the accommodations, location, etc.
https://www.hotel-vendome-nice.com/en/

I booked the Hotel Cronstadt for late May for 180 euros per night. It's the week of the Monaco Grand Prix and the Cannes Film Festival and I had trouble finding anything with availability and a reasonable price. It is only a couple of blocks from the Mediterranean.

Check out the cute Happy Culture hotels. I am staying at Le Grimaldi all next week.

It has been. many years, but we enjoyed the Windsor with its unique rooms (each one has a different artistic theme) and the really lovely garden for breakfast with a small pool); it is an easy walk to the water from there.