i would recommend two excellent hotels :
1. Santa Marina in Ornos
2. Saint John ( St. John beach)
they are both within your price range and their rooms have great seaviews
Even Cavo Tagoo which is one of the top hotels of Mykonos has rooms that meet your criteria....
Cavo Tagoo is located in Tagoo area, and you can walk to the town, as it is not not far away, for the other two hotels you will need either public transportation or a taxi or you could rent a car.

If you have a rental car stay outside of Mykonos Town. Check out the Mykonos Thea Hotel, it's beautiful on a private beach.

www.mykonosthea.com

Many Thanks for your quick response.
Tagoo area looks good from our perspective. How easy is it to take a bus to some good beach in the morning and come back later in the evening? Is the bus frequency good? Also, are car rentals easy in Mykonos?

parking the car at night in Mykonos town isn't easy... but with the exception of August i have always managed to find space... plus there is always the huge public parking next to the old port.....
Busses operate frequently between the town and Platis Yialos, from there you should take a boat to beaches such as Paranga,Paradise or Super Paradise.
Some of these beaches ( Paranga, Paradise) can be reached by bus as well.... or you can go also to other beaches with the bus such as Elia or Kalafatis.....
Ticket is between 1,60 and 1,90 euro.. since Mykonos is a small island ride should take between 15 and 30 min to more remote ones, such as Elia or Kalafatis.
Finally and in regards to hotel Thea, keep in mind please that in whole of Greece there are NO private beaches.
According to the Greek constitution all beaches of Greece are public ( there are only a very few exceptions mainly in Athens)

As for renting a car, this is really no problem.
If you should be on Mykonos during the peakseason , from mid July, i suggest you book in advance, as it can happen that all car rentals are sold out on a short term, especially during weekends.