Welcome to the board...
Try
=SUMPRODUCT(--(MONTH(B2:M2)=1),B3:M3)
However, you may find that this (obviously) doesn't care what year the date is...
So if your dates may contain multiple years, you may get inaccurate results..
If you need to account for the year as well, try
=SUMPRODUCT(--(TEXT(B2:M2,"mmmyyyy")="Jan2011"),B3:M3)