DateTime date = ... // your original date here...
// Don't bother checking, just create a new date for the 1st.
date = new DateTime(date.Year, date.Month, 1);
UPDATE:
The OP has apparently changed the specs:
DateTime date = ... // your original date here...
if (date.Day != 1)
date = new DateTime(date.Year, date.Month, 1).AddMonths(1);
(let the .AddMonths() method worry about the year rolling over in December...)