Brindisi or Bari hotel suggestions

A few years ago I stayed at Oriente in Bari, just close to Teatro Petruzzelli and the favored hotel of opera artists. At breakfast I observed that the mayor of the city had called a meeting there so it should be one of the best around. I actually lived in Bari as a teen for 11 years, but it was the first time I went back. IMHO the important thing is staying north of the railway station; Bari has a medieval area, not dissimilar from an Arab casbah, the 19th century Murat town, laid on a square grid, and the post-war addition south of the railway line. Avoid this one as it is not particularly beautiful and involves a lot of walking around; the Murat city is your best bet and Oriente has probably one of the best locations.
For a following job I was booked at residence zodiacus that actually looks like a B&B. The room, on two levels, was fine, but the breakfast was at a nearby bar and Italian-style (a coffee and a roll).
Both hotels work if you are on public transportation (you will need a taxi from the station to Oriente and you can walk to Zodiacus); there is not much parking space in the centre of Bari. If you have a car then a periferic hotel could be a better choice.

It's been many years since my two stays, but both were B&Bs in the old town of Bari that fit your budget. Le Nicchie has been converted into an apartment rental and was my favorite of the two--it has a nice roof deck. The other place I stayed is Il B&B la Maison del Borgo Antico, but since you will be staying a few days, definitely pick the one that is larger. My husband and I are small people, and we had trouble turning around in the tiny shower. B&Bs in the old town are often converted buildings and very small, but I would endure that to stay in the old town.
I always start my accommodation searches at Booking.com, where you can sort by location, price, and any amenity important to you.

Thank you both for the information.

Recently stayed at the iH Oriente in Bari, mentioned above. Great location, very comfortable hotel, good breakfast, helpful personnel. It may or may not be within your budget depending on when you'll be in Bari.

@Elena thank you for the information. It's is a little high, but it might work out if I can get some hotels at other locations on our trip for less than I thought.