Manuel Maria Castillo is close to the National Palace. The closest hotel I know of would be The Carey House. I stayed there once a little over a year ago. It is a small hotel with small room but very nice and quiet. It is probably within walking distance of where you want to go. Looking at the map on their webpage Moises Garcia street changes name to Manuel Maria Castillo to the east (right) of Dr. Delgado. There is no much to do close by the hotel but the area is safe. You can email them for room rates. I am guessing around $50 USD a night.
One more thing, when staying at the smaller hotels be prepared to pay in cash, USD.
.::Welcome to carey House Hotel::.
Another small hotel I really like is Jaguar Apart/Hotel on Hermanos Delinge. You will need to take a 10 minute $RD150 taxi ride. You can call for a taxi or walk a short block and a half where taxis are always waiting. The hotel does not have a restaurant but there are several with a 5 minute walk. There is a small supermarket less than a 10 minute walk to stock up the frig.
.....:Jaguar Apart Studios::::..........
You also have Hotel Venezia Residence which is less than a 10 minute taxi ride.
located in the heart of Santo Domingo
I have stayed in the above three hotels and all are clean, safe hotels in the $50 - $60 price range. Just depends on what you want.
On the Malecon there are several larger hotels available. InterContinental, Melia, and Jaragua Hotel/Casino but you will easily pay twice or more a night.