Here are some "influential" Senators, off the top of my head:
Bucky Harris
Mickey Vernon
Jim Lemon
Eddie Yost
(above were both Senators and Senator managers. Though Yost only one game. )
Roy Sievers
Camilo Pasqual
Sam Rice
Joe Cronin
Clark Griffith
Expansion Senators:
Frank Howard
Dick Bosman
Ed Brinkman
Chuck Hinton
And of course (as several have mentioned) Walter Johnson
I'd omit Killebrew because he had only two meaningful seasons here.