Try with below query:
declare @from_date datetime;
declare @to_date datetime;
set @to_date = '03/31/2016';
set @from_date = '03/01/2016';
select registration_no, name, category, status, dob
From rpd
where Datepart(month,dob) between datepart(month, @from_date)
and Datepart(month,@to_date)
and (
(Datepart(day,dob) >= datepart(day, @from_date) and datepart(month, dob) = Datepart(month, @from_date))
OR
(Datepart(day,dob) <= datepart(day, @to_date) and datepart(month, dob) = Datepart(month, @to_date))
OR
(Datepart(month,dob) > datepart(month, @from_date) and datepart(month, dob) < Datepart(month, @to_date))
)