The phone number provided by JudyArtley (SHOTGUN) is for the COURT Clerk. Unless the birth occured in Court, there won't be a record for it with the County Clerk. A County Clerk is almost always the COURT clerk.
What you are needing is a BIRTH record, not a Court Record. For births AFTER 1907, the record would be found at the Washington State Department of Health:http://www.doh.wa.gov/EHSPHL/CHS/cert.htm
For the record you seek, in 1903, it "may" be still at the County level, for Douglas County, see:http://www.doh.wa.gov/EHSPHL/CHS/issuing_co.htm
Shotgun, I have sent this information regarding "Court House" searches several times to your postings. Please note WHO does WHAT in the "Court House". A County Recorder, County Assessor, County Heath/Vital Records, are not always in the "Court House".
Also, since Shotgun didn't say, here is the link for the Washington State Archives...you may get lucky:http://www.digitalarchives.wa.gov/default.aspx
Sorry for any Problems encountered.... ...