There was a movie set in Alaska produced in the 1930 or 1940's in
which the heroine says, "Tyler's gone to Sitka." Please tell me the
name of the movie.

Dear snowwhite31,

The movie containing this quote is "Spawn of the North" (1938),
starring George Raft (as Tyler Dawson), Henry Fonda (as Jim Kimmerlee)
and Dorothy Lamour (as Nicky Duval).
Here is a brief plot summary, taken from the Internet Movie Database:

"Two Alaskan salmon fishermen, Tyler Dawson (skipper of the "Who
Cares") and Jim Kimmerlee of the "Old Reliable," are lifelong pals.
Their romantic rivalry over young Dian ends amicably. But a more
serious rift, with violent consequences, arises when Tyler befriends
Russian fish pirates while Jim finds himself aligned with local
vigilantes."

Source:
IMDb: Plot Summary for Spawn of the North (1938), by Rod Crawford
http://us.imdb.com/Plot?0030775

Please follow this link to view additional information on this movie:

IMDb: Spawn of the North (1938)
http://us.imdb.com/Title?0030775

"Spawn of the North" is currently available as VHS in American NTSC
format from Amazon.com for $14.98:
http://www.amazon.com/exec/obidos/ASIN/0783224478/ref=ase_imdb-adbox/104-5259881-1876733

Search strategy: I vaguely remembered seeing a classic movie set in
Alaska where Sitka, the old capital of Russian Alaska was mentioned in
the way you quoted. To find out more, I systematically searched the
Internet Movie Database for all movies made between 1930 and 1949,
playing in Alaska and having a character named Tyler. There is only
one film matching all points, and when I read the summary, I knew it
is the right one.

Hope this answers your question!
Best regards,
Scriptor

This was really wonderful. At that time I was about seven or eight
years old, and my mother would quote this phrase. I know that she
probably did not see the movie, but the words had stuck in my mind.
Now I know how to search better on the internet. Many thanks.